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GROWTH  AND  FORM 


Ah  instantdnedus  photograph  of  a  'splash'  of  milk.  From  Harold  E. 
'   '"*'       KdgertbTiTirlassachusetts  Institute  of  Technology 


See  p.  390 


ON  94 

GROWTH  AND  FORM  ^^ 


BY 


D'ARCY  WENTWORTH  THOMPSON 


A  new  edition 

MARINE 

BIOLOGICAL 

LABORATORY 

LIBRARY 

WOODS  HOLE,  MASS. 
W.  H.  0.  1. 

CAMBRIDGE:  AT  THE  UNIVERSIT 
NEW  YORK:  THE  MACMILLAN  CC 

Y  PRESS 
)MPANY 

1945 

"The  reasonings  about  the  wonderful  and  intricate  operations 
of  Nature  are  so  full  of  uncertainty,  that,  as  the  Wise-man  truly 
observes,  hardly  do  we  guess  aright  at  the  things  jhat  are  upon 
earth,  and  with  labour  do  we  find  the  things  that  are  before  us.'' 
Stephen  Hales,  Vegetable  Staticks  (1727),  p.  318,  1738. 

"Ever  since  I  have  been  enquiring  into  the  works  of  Nature 
I  have  always  loved  and  admired  the  Simplicity  of  her  Ways." 
Dr  George  Martine  (a  pupil  of  Boorhaave's),  in  Medical  Essays  and 
Observations,  Edinburgh,  1747. 


PREFATORY  NOTE 

THIS  book  of  mine  has  little  need  of  preface,  for  indeed  it  is 
"all  preface"  from  beginning  to  end.  I  have  written  it  as 
an  easy  introduction  to  the  study  of  organic  Form,  by  methods 
which  are  the  common-places  of  physical  science,  which  are  by 
no  means  novel  in  their  application  to  natural  history,  but  which 
nevertheless  naturalists  are  little  accustomed  to  employ. 

It  is  not  the  biologist  with  an  inkling  of  mathematics,  but 
the  skilled  and  learned  mathematician  who  must  ultimately  deal 
with  such  problems  as  are  sketched  and  adumbrated  here.  I  pretend 
to  no  mathematical  skill,  but  I  have  made  what  use  I  could  of 
what  tools  I  had;  I  have  dealt  with  simple  cases,  and  the  mathe- 
matical methods  which  I  have  introduced  are  of  the  easiest  and 
simplest  kind.  Elementary  as  they  are,  my  book  has  not  been 
written  without  the  help — the  indispensable  help^ — of  many  friends. 
Like  Mr  Pope  translating  Homer,  when  I  felt  myself  deficient  I 
sought  assistance!  And  the  experience  which  Johnson  attributed 
to  Pope  has  been  mine  also,  that  men  of  learning  did  not  refuse 
to  help  me. 

I  wrote  this  book  in  wartime,  and  its  revision  has  employed 
me  during  another  war.  It  gave  me  solace  and  occupation,  when 
service  was  debarred  me  by  my  years. 

Few  are  left  of  the  friends  who  helped  me  write  it,  but  I  do  not 
forget  the  debt  I  owe  them  all.  Let  me  add  another  to  these 
kindly  names,  that  of  Dr  G.  T.  Bennett,  of  Emmanuel  College, 
Cambridge;  he  has  never  wearied  of  collaboration  with  me,  and 
his  criticisms  have  been  an  education  to  receive. 

D.  W.  T. 
1916-1941. 


American   edition   published August,  1942 

Reprinted    January,  1943 

Reprinted    May,  1944 

Reprinted    May,  1945 


PRINTED    IN    THE    UNITED    STATES    OF    AMERICA 


CONTENTS 

CHAP.  PAGE 

I.  Introductory 1 

II.  On  Magnitude 22 

III.  The  Rate  of  Growth 78 

IV.  On  the  Internal  Form  and  Structure  of  the  Cell    .         .  286 

V.  The  Forms  of  Cells 346 

VI.  A  Note  on  Adsorption 444 

VII.  The  Forms  of  Tissues,  or  Cell-aggregates          .        .        .  465 

VIII.  The  same  (continued) .         .  566 

IX.  On  Concretions,  Spicules,  and  Spicular  Skeletons   .        .  645 

X.  A  Parenthetic  Note  on  Geodetics 741 

XI.  The  Equiangular  Spiral 748 

XII.      The  Spiral  Shells  of  the  Foraminifera      ....  850 

XIII.  The  Shapes  of  Horns,   and   of  Teeth   or  Tusks:  with 

a  Note  on  Torsion 874 

XIV.  On  Leaf-arrangement,  or  Phyllotaxis       ....  912 

XV.  On  the  Shapes  of  Eggs,  and  of  certain  other  Hollow 

Structures 934, 

XVI.  On  Form  and  Mechanical  Efficiency           ....  958 

XVII.  On  the  Theory  of  Transformations,  or  the  Comparison 

OF  Related  Forms                    1026 

Epilogue 1093 

Index 1095 

Plates 

A  Splash  of  Milk Frontispiece 

The  Latter  Phase  of  a  Splash         .         .  "      .         .         .         .    facing  page  390 


"The  mathematicians  are  well  acquainted  with  the  difference 
between  pure  science,  which  has  to  do  only  with  ideas,  and  the 
application  of  its  laws  to  the  use  of  life,  in  which  they  are  con- 
strained to  submit  to  the  imperfections  of  matter  and  the  influence 
of  accident."  Dr  Johnson,  in  the  fourteenth  Rambler,  May  5,  1750. 

"Natural  History. .  .is  either  the  beginning  or  the  end  of  physical 
science."  Sir  John  Herschel,  in  The  Study  of  Natvml  Philosophy, 
p.  221,  1831. 

"I  believe  the  day  must  come  when  the  biologist  will — without 
being  a  mathematician — not  hesitate  to  use  mathematical  analysis 
when  he  requires  it."     Karl  Pearsonf  in  Nature,  January  17,  1901. 


CHAPTER  I 

INTRODUCTORY 

Of  the  chemistry  of  his  day  and  generation,  Kant  declared  that  it 
was  a  science,  but  not  Science — eine  Wissenschaft,  aber  nicht  Wissen- 
schaft — for  that  the  criterion  of  true  science  lay  in  its  relation  to 
mathematics*.  This  was  an  old  story :  for  Roger  Bacon  had  called 
mathematics  porta  et  clavis  scientiarum,  and  Leonardo  da  Vinci  had 
said  much  the  samef.  Once  again,  a  hundred  years  after  Kant, 
Du  Bois  Reymond,  profound  student  of  the  many  sciences  on  which 
physiology  is  based,  recalled  the  old  saying,  and  declared  that 
chemistry  would  only  reach  the  rank  of  science,  in  the  high  and 
strict  sense,  when  it  should  be  found  possible  to  explain  che^dcal 
reactions  in  the  light  of  their  causal  relations  to  the  velocities, 
tensions  and  conditions  of  equihbrium  of  the  constituent  molecules ; 
that,  in  short,  the  chemistry  of  the  future  must  deal  with  molecular 
mechanics  by  the  methods  and  in  the  strict  language  of  mathematics, 
as  the  astronomy  of  Newtoii  and  Laplace  dealt  with  the  stars  in 
their  courses.  We  know  how  great  a  step  was  made  towards  this 
distant  goal  as  Kant  defined  it,  when  van't  Hoif  laid  the  firm 
foundations  of  a  mathematical  chemistry,  and  earned  his  proud 
epitaph — Physicam  chemiae  adiunxitX- 

We  need  not  wait  for  the  full  reahsation  of  Kant's  desire,  to  apply 
to  the  natural  sciences  the  principle  which  he  laid  down.  Though 
chemistry  fall  short  of  its  ultimate  goal  in  mathematical  mechanics  §, 
nevertheless  physiology  is   vastly  strengthened   and   enlarged   by 

*  "  Ich  behaupte  nur  dass  in  jeder  besonderen  Naturlehre  nur  so  viel  eigentliche 
Wissenschaft  angetroffen  konne  als  darin  Mathematik  anzutreffen  ist" :  Gesammelte 
Schriften,  iv,  p.  470. 

t  "Nessuna  humana  investigazione  si  puo  dimandare  vera  scienzia  s'essa  non 
passa  per  le  matematiche  dimostrazione." 

X  Cf.  also  Crum  Brown,  On  an  application  of  Mathematics  to  Chemistry,  Trans. 
R.S.E.  XXIV,  pp.  691-700,  1867. 

§  Ultimate,  for,  as  Francis  Bacon  tells  us:  Mathesis  philosophiam  naturalem 
terminare  debet,  non  generare  aut  procreare. 


2  INTRODUCTORY  <^i,J^^H 

making  use  of  the  chemistry,  and  of  the  physics,  of  the  age.  Littl 
by  httle  it  draws  nearer  to  our  conception  of  a  true  science  with 
each  branch  of  physical  science  which  it  brings  into  relation  with 
itself:  with  every  physical  law  and  mathematical  theorem  which  it 
learns  to  take  into  its  employ*.  Between  the  physiology  of  Haller, 
fine  as  it  was,  and  that  of  Liebig,  Helmholtz,  Ludwig,  Claude 
Bernard,  there  was  all  the  difference  in  the  worldf. 

As  soon  as  we  adventure  on  the  paths  of  the  physicist,  we  learn 
to  weigh  and  to  measure,  to  deal  with  time  and  space  and  mass  and 
their  related  concepts,  and  to  find  more  and  more  our  knowledge 
expressed  and  our  needs  satisfied  through  the  concept  of  number, 
as  in  the  dreams  and  visions  of  Plato  and  Pythagoras ;  for  modern 
chemistry  would  have  gladdened  the  hearts  of  those  great  philo- 
sophic dreamers.  Dreams  apart,  numerical  precision  is  the  very 
soul  of  science,  and  its  attainment  affords  the  best,  perhaps,  the 
only  criterion  of  the  truth  of  theories  and  the  correctness  of  experi- 
mentsj:.  So  said  Sir  John  Herschel,  a  hundred  years  ago;  and 
Kant  had  said  that  it  was  Nature  herself,  and  not  the  mathematician, 
who  brings  mathematics  into  natural  philosophy. 

But  the  zoologist  or  morphologist  has  been  slow,  where  the 
physiologist  has  long  been  eager,  to  invoke  the  aid  of  the  physical 
or  mathematical  sciences;  and  the  reasons  for  this  difference  lie 
deep,  and  are  partly  rooted  in  old  tradition  and  partly  in  the 
diverse  minds  and  temperaments  of  men.  To  treat  the  living  body 
as  a  mechanism  was  repugnant,  and  seemed  even  ludicrous,  to 
Pascal  §;  and  Goethe,  lover  of  nature  as  he  was,  ruled  mathematics 
out  of  place  in  natural  history.  Even  now  the  zoologist  has  scarce 
begun  to  dream  of  defining  in  mathematical  language  even  the 
simplest  organic  forms.     When  he  meets  with  a  simple  geometrical 

*  "Sine  profunda  Mechanices  Scientia  nil  veri  vos  intellecturos,  nil  boni  pro- 
laturos  aliis":   Boerhaave,  De  usu  ratiocinii  Mechanici  in  Medicina,  1713. 

t  It  is  well  within  my  own  memor  how  Thomson  and  Tait,  and  Klein  and 
Sylvester  had  to  lay  stress  on  the  mathematical  aspect,  and  urge  the  mathematical 
study,  of  physical  science  itself! 

X  Dr  Johnson  says  that  "to  count  is  a  modern  practice,  the  ancient  method  was 
to  guess";  but  Seneca  was  alive  to  the  difference — "magnum  esse  solem  philosophus 
probabit,  quantus  sit  mathematicus." 

§  Cf.  Pensees,  xxix,  "II  faut  dire,  en  gros,  cela  se  fait  par  figure  et  mouvement, 
car  cela  est  yrai.  Mais  de  dire  quels,  et  composer  la  machine,  cela  est  ridicule, 
car  cela  est  inutile,  et  incertain,  et  penible." 


I]  OF  ADAPTATION  AND  FITNESS  3 

construction,  for  instance  in  the  honeycomb,  he  would  fain  refer  it 
to  psychical  instinct,  or  to  skill  and  ingenuity,  rather  than  to  the 
operation  of  physical  forces  or  mathematical  laws;  when  he  sees  in 
snail,  or  nautilus,  or  tiny  foraminiferal  or  radiolarian  shell  a  close 
approach  to  sphere  or  spiral,  he  is  prone  of  old  habit  to  believe  that 
after  all  it  is  something  more  than  a  spiral  or  a  sphere,  and  that  in 
this  "something  more"  there  lies  what  neither  mathematics  nor 
physics  can  explain.  In  short,  he  is  deeply  reluctant  to  compare 
the  living  with  the  dead,  or  to  explain  by  geometry  or  by  mechanics 
thp  things  which  have  their  part  in  the  mystery  of  life.  Moreover 
he  IS  httle  inclined  to  feel  the  need  of  such  explanations,  or  of  such 
extension  of  his  field  of  thought.  He  is  not  without  some  justifi- 
cation if  he  feels  that  in  admiration  of  nature's  handiwork  he  has 
an  horizon  open  before  his  eyes  as  wide  as  any  man  requires.  He 
has  the  help  of  many  fascinating  theories  within  the  bounds  of  his 
own  science,  which,  though  a  little  lacking  in  precision,  serve  the 
purpose  of  ordering  his  thoughts  and  of  suggesting  new  objects  of 
enquiry.  His  art  of  classification  becomes  an  endless  search  after 
the  blood-relationships  of  things  living  and  the  pedigrees  of  things 
dead  and  gone.  The  facts  of  embryology  record  for  him  (as  Wolff, 
von  Baer  and  Fritz  Miiller  proclaimed)  not  only  the  life-history  of 
the  individual  but  the  ancient  annals  of  its  race.  The  facts  of 
geographical  distribution  or  even  of  the  migration  of  birds  lead  on 
and  on  to  speculations  regarding  lost  continents,  sunken  islands,  or 
bridges  across  ancient  seas.  Every  nesting  bird,  every  ant-hill  or 
spider's  web,  displays  its  psychological  problenis  of  instinct  or  intel- 
ligence. Above  all,  in  things  both  great  and  small,  the  naturajist 
is  rightfully  impressed  and  finally  engrossed  by  the  peculiar  beauty 
which  is  manifested  in  apparent  fitness  or  "adaptation" — the  flower 
for  the  bee,  the  berry  for  the  bird. 

Some  lofty  concepts,  like  space  and  number,  involve  truths  remote 
from  the  category  of  causation;  and  here  we  must  be  content,  as 
Aristotle  says,  if  the  mere  facts  be  known*.  But  natural  history 
deals  with  ephemeral  and  accidental,   not  eternal  nor  universal 

*  ovK  diraiTrjTeop  5  ov8i  ttju  airiav  ofxoiojs,  dW  LKavou  ^v  tlctl  rb  on  deLxdrjvai  /caXcDj 
Eth.  Nic.  1098a,  33.  Teleologist  as  he  was  at  heart,  Aristotle  realised  that  mathematics 
was  on  another  plane  to  teleology:  rds  5^  fxad-qfiariKas  oOdeva  iroieladai  \6you  wepi 
dyaduv  xat  KaKQv.     Met.  996a,  35. 


4  INTRODUCTORY  [ch. 

things ;  their  causes  and  effects  thru^  themselves  on  our  curiosity,  and 
become  the  ultimate  relations  to  which  our  contemplation  extends*. 

Time  out  of  mind  it  has  been  by  way  of  the  "final  cause,"  by  the 
teleological  concept  of  end,  of  purpose  or  of  "design,"  in  one  of  its 
many  forms  (for  its  moods  are  many),  that  men  have  been  chiefly 
wont  to  explain  the  phenomena  of  th«  Hving  world ;  and  it  will  be 
so  while  men  have  eyes  to  see  and  ears  to  hear  withal.  With  Galen, 
as  with  Aristotlef,  it  was  the  physician's  way;  with  John  Ray  J,  as 
with  Aristotle,  it  was  the  naturahst's  way;  with  Kant,  as  with 
Aristotle,  it  was  the  philosopher's  way.  It  was  the  old  Hebrew 
way,  and  has  its  splendid  setting  in  the  story  that  God  made  "every 
plant  of  the  field  before  it  was  in  the  earth,  and  every  herb  of  the 
field  before  it  grew."  It  is  a  common  way,  and  a  great  way;  for  it 
brings  with  it  a  glimpse  of  a  great  vision,  and  it  hes  deep  as  the 
love  of  nature  in  the  hearts  of  men. 

The  argument  of  the  final  cause  is  conspicuous  in  eighteenth- 
century  physics,  half  overshadowing  the  "efficient"  or  physical 
cause  in  the  hands  of  such  men  as  Euler§,  or  Fermat  or  Maupertuis, 
to  whom  Leibniz  1 1  had  passed  it  on.  Half  overshadowed  by  the 
mechanical  concept,  it  runs  through  Claude  Bernard's  Legons  sur  les 
phenomenes  de  la  Vie^,  and  abides  in  much  of  modern  physiology**. 

*  "All  reasonings  concerning  matters  of  fact  seem  to  be  founded  on  the  relation 
of  Cause  and  Effect.  By  means  of  that  relation  alone  we  go  beyond  the  evidence 
of  our  memory  and  senses":   David  Hume,  On  the  Operations  of  the  Understanding. 

t  E.g.  "In  the  works  of  Nature  purpose,  not  accident,  is  the  main  thing":  to  yap 
fir)  TvxofTWs,  d\\'  eveKOL  rtros,  ev  tols  tt]S  (pixreajs  ^pyoLs  ecrl  ko-I  jxaXicra.    PA,  645a,  24. 

X  E.g.  "Quaeri  fortasse  a  nonnullis  potest,  Quis  Papilionum  usus?  Respondeo, 
ad  ornatum  Universi,  et  ut  hominibus  spectaculo  sint."  Joh.  Rail,  Hist.  Insedorum, 
p.  109. 

§  "Quum  enim  Mundi  universi  fabrica  sit  perfectissima,  atque  a  Creators 
sapientissimo  absoluta,  nihil  omnino  in  Mundo  contingit  in  quo  non  maximi 
minimive  ratio  quaepiam  eluceat;  quamobrem  dubium  prorsus  est  nullum  quin 
omnes  Mundi  effectus  ex  causis  finalibus,  ope  Methodi  maximorum  et  minimorum, 
aeque  feliciter  determinari  queant  atque  ex  ipsis  causis  efficientibus."  Methodus 
inveniendi,  etc.,  1744,  p.  24o  {cit.  Mach,  Science  of  Mechanics,  1902,  p.  455). 

!|  Cf.  Opera  (ed.  Erdmann),  p.  106,  "Bien  loin  d'exclure  les  causes  finales... 
c'est  de  la  qu'il  faut  tout  deduire  en  Physique":  in  sharp  contrast  to  Descartes's 
teaching,  "  NuUas  unquam  res  naturales  a  fine,  quem  Deus  aut  Natura  in  iis  faciendis 
sib  jproposuit,  desumemus,  etc."   Princip.  i,  28. 

Tj  Cf.  p.  162.  "La  force  vitale  dirige  des  phenomenes  qu'elle  ne  produit  pas: 
les  agents  physiques  produisent  des  phenomenes  qu'ils  ne  dirigent  pas." 

**  It  is  now  and  then  conceded  with  reluctance.  Thus  Paolo  Enriques,  a  learned 
and  philosophic  naturalist,  writing  '"dell' economia  di  sostanza  nelle  osse  cave" 


I]  OF  THE  FINAL  CAUSE  5 

Inherited  from  Hegel,  it  dominated  Oken's  Naturphilosophie  and 

lingered  among  his  later  disciples,  who  were  wont  to  liken  the  course 

of  organic  evolution  not  to  the  straggling  branches  of  a  tree,  but  to 

the  building  of  a  temple,  divinely  planned,  and  the  crowning  of  it 

with  its  pohshed  minarets*. 

It  is  retained,  somewhat  crudely,  in  modern  embryology,  by  those 

who  see  in  th^  early  processes  of  growth  a  significance  ''rather 

prospective  than  retrospective,"  such  that  the  enlbryonic  phenomena 

must  "be  referred  directly  to  their  usefulness  in  building  up  the 

body  of  the  future  animalf": — which  is  no  more,  and  no  less,  than 

to  say,  with  Aristotle,  that  the  organism  is  the  reAo?,  or  final  cause, 

of  its  own  processes  of  generation  and  development.     It  is  writ 

large  in  that  Entelechyij:  which  Driesch  rediscovered,  and  which  he 

made  known  to  many  who  had  neither  learned  of  it  from  Aristotle, 

nor  studied  it  with  Leibniz,  nor  laughed  at  it  with  Rabelais  and 

Voltaire.     And,  though  it  is  in  a  very  curious  way,   we  are  told 

that  teleology  was   "refounded,   reformed  and  rehabihtated "   by 

Darwin's  concept  of  the  origin  of  species §;    for,  just  as  the  older 

naturalists  held  (as  Addison ||  puts  it)  that  "the  make  of  every  kind 

of  animal  is  different  from  that  of  every  other  kind;   and  ye.  ti^ere 

is  not  the  least  turn  in  the  muscles,  or  twist  in  the  fibres  of  any  one, 

which  does  not  render  them  more  proper  for  that  particular  animal's 

way  of  life  than  any  other  cut  or  texture  of  them  would  have  been  " : 

so,  by  the  theory  of  natural  selection,  "every  variety  of  form  and 

colour  was  urgently  and  absolutely  called  upon  to  produce  its  title 

{Arch.  f.  Entw.  Mech.  xx,  1906),  says  "una  certa  impronta  di  teleologismo  qua 
e  la  e  rimasta,  mio  malgrado,  in  questo  scritto." 

*  Cf.  John  Cleland,  On  terminal  forms  of  life,  Journ.  Anat.  and  Physiol. 
XVIII,  1884. 

t  Conklin,  Embryology  of  Crepidula,  Journ.  of  Morphol.  xin,  p.  203,  1897; 
cf.  F.  R.  Lillie,  Adaptation  in  cleavage.  Wood's  Hole  Biol.  Lectures,  1899,  pp.  43-67. 

X  I  am  inclined  to  trace  back  Driesch's  teaching  of  Entelechy  to  no  less  a  person 
than  Melanchthon.  When  Bacon  {de  Augm.  iv,  3)  states  with  disapproval  that 
the  soul  "has  been  regarded  rather  as  a  function  than  as  a  substance,"  Leslie 
Ellis  points  out  that  he  is  referring  to  Melanchthon's  exposition  of  the  Aristotelian 
doctrine.  For  Melanchthon,  whose  view  of  the  peripatetic  philosophy  had  great 
and  lasting  influence  in  the  Protestant  Universities,  affirmed  that,  according  to 
the  true  view  of  Aristotle's  opinion,  the  soul  is  not  a  substance  but  an  ivreX^xeia,  or 
function.  He  defined  it  as  Sufa/xis  quaedam  ciens  actiones — a  description  all  but 
identical  with  that  of  Claude  Bernard's  '''force  vitale.'' 

§   Ray  Lankester,  art.  Zoology,  Encycl.  Brit.  (9th  edit.),  1888,  p.  806. 

!l   Spectator,  No.  120. 


6  INTRODUCTORY  [ch. 

to  existence  either  as  an  active  useful  agent,  or  as  a  survival"  of 
such  active  usefulness  in  the  past.  But  in  this  last,  and  very 
important  case,  we  have  reached  a  teleology  without  a  rdXos,  as 
men  like  Butler  and  Janet  have  been  prompt  to  shew,  an  "adapta- 
tion" without  "design,"  a  teleology  in  which  the  final  cause  becomes 
little  more,  if  anything,  than  the  mere  expression  or  resultant  of  a 
sifting  out  of  the  good  from  the  bad,  or  of  the  better  from  the  worse, 
in  short  of  a  process  of  mechanism.  The  apparent  manifestations 
of  purpose  or  adaptation  become  part  of  a  mechanical  philosophy, 
"une  forme  methodologique  de  connaissance*,"  according  to  which 
"la  Nature  agit  tou jours  par  les  moyens  les  plus  simplest,"  and 
"chaque  chose  finit  toujours  par  s'accommoder  a  son  miUeu,"  as  in 
the  Epicurean  creed  or  aphorism  that  ^atnTe  finds  a  use  for  every- 
thing J.  In  short,  by  a  road  which  resembles  but  is  not  the  same  as 
Maupertuis's  road,  we  find  our  way  to  the  very  world  in  which  we 
are  living,  and  find  that,  if  it  be  not,  it  is  ever  tending  to  become, 
"the  best  of  all  possible  worlds §." 

But  the  use  of  the  teleological  principle  is  but  one  way,  not  the 
whole  or  the  only  way,  by  which  we  may  seek  to  learn  how  things 
came  to  be,  and  to  take  their  places  in  the  harmonious  complexity 
of  the  world.  To  seek  not  for  end^  but  for  antecedents  is  the  way 
of  the  physicist,  who  finds  "causes"  in  what  he  has  learned  to 
recognise  as  fundamental  properties,  or  inseparable  concomitants, 
or  unchanging  laws,  of  matter  and  of  energy.  In  Aristotle's  parable, 
the  house  is  there  that  men  may  live  in  it ;  but  it  is  also  there  because 
the  builders  have  laid  one  stone  upon  another.  It  is  as  a  mechanism, 
or  a  mechanical  construction,  that  the  physicist  looks  upon  the 
world;  and  Democritus,  first  of  physicists  and  one  of  the  greatest 
of  the  Greeks,  chose  to  refer  all  natural  phenomena  to  mechanism 
and  set  the  final  cause  aside. 

*  So  Newton,  in  the  Preface  to  the  Principia:  "Natura  enim  simplex  est,  et 
rerura  causis  superfluis  non -luxuriat";  "Nature  is  pleased  with  simplicity,  and 
affects  not  the  pomp  of  superfluous  causes."  Modern  physics  finds  the  perfection 
of  mathematical  beauty  in  what  Newton  called  the  perfection  of  simplicity. 

t  Janet,  Les  Causes  Finales,  1876,  p.  350. 

X  "Nil  ideo  quoniam  natumst  in  corpore  ut  uti  Possemus  sed  quod  natumst  id 
procreat  usum."     Lucret.  iv,  834. 

§  The  phrase  is  Leibniz's,  in  his  Theodicee:  and  harks  back  to  Aristotle — If  one 
way  be  better  than  another,  that  you  may  be  sure  is  Nature's  way;  Nic.  Eth. 
10996,  23  et  al. 


I]  OF  EFFICIENT  AND  FINAL  CAUSES  7 

Still,  all  the  whiLe,  like  warp  and  woof,  mechanism  and  teleology 
are  interwoven  together,  and  we  must  not  cleave  to  the  one  nor 
despise  the  other;  for  their  union  is  rooted  in  the  very  nature  of 
totality.  We  may  grow  shy  or  weary  of  looking  to  a  final  cause 
for  an  explanation  of  our  phenomena ;  but  after  we  have  accounted 
for  these  on  the  plainest  principles  of  mechanical  causation  it  may 
be  useful  and  appropriate  to  see  how  the  final  cause  would  tally 
wuth  the  other,  and  lead  towards  the  same  conclusion*.  Maupertuis 
had  Uttle  hking  for  the  final  cause,  and  shewed  some  sympathy  with 
Descartes  in  his  repugnance  to  its  appHcation  to  physical  science. 
But  he  found  at  last,  taking  the  final  and  the  efficient  causes  one  with 
another,  that  "I'harmonie  de  ces  deux  attributs  est  si  parfaite  que 
sans  doute  tous  les  effets  de  la  Nature  se  pourroient  deduire  de 
chacun  pris  separement.  Une  Mecanique  aveugle  et  necessaire  suit 
les  dessins  de  I'lntelHgence  la  plus  eclairee  et  la  plus  hbref."  Boyle 
also,  the  Father  of  Chemistry,  wrote,  in  his  latter  years,  a  Disquisition 
about  the  Final  Causes  of  Natural  Things:  Wherein  it  is  Inquired 
Whether,  And  {if  at  all)  With  what  Cautions,  a  Naturalist  should  admit 
Themi  He  found  "that  all  consideration  of  final  cause  is  not  to  be 
banished  from- Natural  Philosophy...";  but  on  the  other  hand 
''that  the  naturahst  who  would  deserve  that  name  must  not  let 
the  search  and  knowledge  of  final  causes  make  him  neglect  the  in- 
dustrious indagation  of  efficients  J."  In  our  own  day  the  philosopher 
neither  minimises  nor  unduly  magnifies  the  mechanical  aspect  of 
the  Cosmos;  nor  need  the  naturahst  either  exaggerate  •  or  be- 
little the  mechanical  phenomena  which  are  profoundly  associated 
with  Life,  and  inseparable  from  our  understanding  of  Growth  and 
Form.  ^ 

*  "S'il  est  dangereux  de  se  servir  des  causes  finales  a  priori  pour  trouver  les  lois 
des  phenomenes,  11  est  peut-etre  utile  et  il  est  au  moins  curieux  de  faire  voir  com- 
ment le  principe  des  causes  finales  s'accorde  avec  les  lois  des  phenomenes,  pourvu 
qu'on  commence  par  determiner  ces  lois  d'apres  les  principes  de  mecanique  clairs 
et  incontestables."  (D'Aiembert,  Art.  Causes  finales,  Encyclopedie,  ii,  p.  789,  1751.) 
I  SeeJiis  essay  on  the  '''Accord  des  differentes  lois  de  la  Nature." 
X  Cf.  also  Leibniz  {Discours  de  la  Metaphysique:  Lettres  inedites,  ed.  de  Careil, 
1857,  p.  354),  "L'un  et  I'autre  est  bon,  I'un  et  I'autre  peut  etre  utile... et  les 
auteurs  qui  suivent  ces  deux  routes  differentes  ne  devraient  pas  se  maltraiter." 
Or  again  in  the  Monadologie,  "Les  ames  agissent  selon  les  causes  finales. ...  Les 
corps  agissent  selon  les  lois  des  causes  efficientes  ou  des  mouveraents.  Et  les 
deux  regnes,  celui  des  causes  efficientes  et  des  causes  finales  sont  harmonieux 
entre  eux." 


8  INTRODUCTORY  [ch. 

Nevertheless,  when  philosophy  bids  us  hearken  and  obey  the 
lessons  both  of  mechanical  and  of  teleological  interpretation,  the 
precept  is  hard  to  follow:  so  that  oftentimes  it  has  come  to  pass, 
just  as  in  Bacon's  day,  that  a  leaning  to  the  side  of  the  final  cause 
"hath  intercepted  the  severe  and  diligent  enquiry  of  all  real  and 
physical  causes,"  and  has  brought  it  about  that  "the  search  of  the 
physical  cause  hath  been  neglected  and  passed  in  silence."  So  long 
and  so  far  as  "fortuitous  variation*"  and  the  "survival  of  the 
fittest"  remain  engrained  as  fundamental  and  satisfactory  hypo- 
theses in  the  philosophy  of  biology,  so  long  will  these  "satisfactory 
and  specious  causes"  tend  to  stay  "severe  and  diligent  enquiry. . . 
to  the  great  arrest  and  prejudice  of  future  discovery."  Long 
before  the  great  Lord  Keeper  wrote  these  words,  Roger  Bacon  had 
shewn  how  easy  it  is,  and  how  vain,  to  survey  the  operations  of 
Nature  and  idly  refer  her  wondrous  works  to  chance  or  accident, 
or  to  the  immediate  interposition  of  Godf. 

The  difiiculties  which  surround  the  concept  of  ultimate  or  "real" 
caugation,  in  Bacon's  or  Newton's  sense  of  the  word,  the  in- 
superable difficulty  of  giving  any  just  and  tenable  account  of  the 
relation  of  cause  and  effect  from  the  empirical  point  of  view,  need 
scarcely  hinder  us  in  our  physical  enquiry.  As  students  of  mathe- 
matical and  experimental  physics  we  are  content  to  deal  with  those 
antecedents,  or  concomitants,  of  our  phenomena  without  which  the 
phenomenon  does  not  occur — with  causes,  in  short,  which,  aliae  ex 
aliis  aptaetet  necessitate' nexae,  are  no  more,  and  no  less,  than  con- 
ditions sine  qua  non.  Our  purpose  is  still  adequately .  fulfilled : 
inasmuch  as  we  are  still  enabled  to  correlate,  and  to  equate,  our 
particular  phenomena  with  more  and  more  of  the  physical  phenomena 
around,  and  so  to  weave  a  web  of  connection  and  interdependence 
which  shall  serve  our  turn,  though  the  metaphysician  withhold  from 
that  interdependence  the  title  of  causality  J.     We  come  in  touch 

*  The  reader  will  understand  that  I  speak,  not  of  the  "severe  and  diligent 
enquiry"  of  variation  or  of  fortuity,  but  merely  of  the  easy  assumption  that  these 
phenomena  are  a  sufficient  basis  on  which  to  rest,  with  the  all-powerful  help  of 
natural  selection,  a  theory  of  definite  and  progressive  evolution. 

f  Op.  tert.  (ed.  Brewer,  p.  99).  "Ideo  mirabiles  actiones  naturae,  quae  tota 
die  fiunt  in  nobis  et  in  rebus  coram  oculis  nostris,  non  percipimus;  sed  aestimamus 
eas  fieri  vel  per  specialem  operationem  divinam. .  .vel  a  casu  et  fortuna," 

t  Cf.  Fourier's  phrase,  in  his  Theorie  de  la  Chaleur,  with  which  Thomson  and 
Tait  prefaced  their  Treatise  an  Natural  Philosophy:  "Les  causes  primordiales  ne 


I]  OF  ULTIMATE  CAUSATION  9 

with  what  the  schoohnen  called  a  ratio  cognoscendi,  though  the  true 
ratio  efflciendi  is  still  enwrapped  in  many  mysteries.  And  so  handled, 
the  quest  of  physical  causes  merges  with  another  great  Aristotelian 
theme — the  search  for  relations  between  things  apparently  dis- 
connected, and  for  "similitude  in  things  to  common  view  unlike*." 
Newton  did  not  shew  the  cause  of  the  apple  falling,  but  he  shewed 
a  simihtude  ("the  more  to  increase  our  wonder,  with  an  apple") 
between  the  apple  and  the  starsf.  By  doing  so  he  turned  old  facts 
into  new  knowledge ;  and  was  well  content  if  he  could  bring  diverse 
phenomena  under  "two  or  three  Principles  of  Motion"  even  "though 
the  Causes  of  these  Principles  were  not  yet  discovered". 

Moreover,  the  naturalist  and  the  physicist  will  continue  to  speak 
of  "causes",  just  as  of  old,  though  it  may  be  with  some  mental 
reservations :  for,  as  a  French  philosopher  said  in  a  kindred  difficulty : 
"ce  sont  la  des  manieres  de  s'exprimer,  et  si  elles  sont  interdites 
il  faut  renoncer  a  parler  de  ces  choses." 

The  search  for  differences  or  fundamental  contrasts  between  the 
phenomena  of  organic  and  inorganic,  of  animate  and  inanimate, 
things,,  has  occupied  many  men's  minds,  while  the  search  for  com- 
munity of  principles  or  essential  simihtudes  has  been  pursued  by 
few;  and  the  contrasts  are  apt  to  loom  too  large,  great  though  they 
may  be.  M.  Dunan,  discussing  the  Probleme  de  la  Viel,  in  an  essay 
which  M.  Bergson  greatly  commends,  declares  that  "les  lois  physico- 
chimiques  sont  aveugles  et  brutales ;  la  ou  elles  regnent  seules,  au 
lieu  d'un  ordre  et  d'un  concert,  il  ne  pent  y  avoir  qu'incoherence  et 
chaos."  But  the  physicist  proclaims  aloud  that  the  physical 
phenomena  which  meet  us  by  the  way  have  their  forms  not  less 
beautiful  and  scarce  less  varied  than  those  which  move  us  to  admira- 

nous  sont  point  connues;  mais  elles  sont  assujetties  a  des  lois  simples  et  eonstantes, 
que  Ton  peut  decouvrir  par  I'observation,  et  dont  I'etude  est  I'objet  de  la  philosophie 
naturelle." 

*  "Plurimum  amo  analogias,  fidelissimos  meos  magistros,  omnium  Naturae 
arcanorum  conscios,"  said  Kepler;  and  Perrin  speaks  with  admiration,  in  Les 
Atonies,  of  men  like  Galileo  and  Carnot,  who  "possessed  the  power  of  perceiving 
analogies  to  an  extraordinary  degree."  Hure^e  declared,  and  Mill  said  much  the 
same  thing,  that  all  reasoning  whatsoever  depends  on  resemblance  or  analogy, 
and  the  power  to  recognise  it.  Comparative  anatomy  (as  Vicq  d'Azyr  first  called 
it),  or  comparative  physics  (to  use  a  phrase  of  Mach's),  are  particular  instances  of 
a  sustained  search  for  analogy  or  similitude. 

t  As  for  Newton's  apple,  see  De  Morgan,  in  Notes  and  Queries  (2),  vi,  p.  169,  1858. 

I  Revue  Philosophique,  xxxiii,  1892. 


10  INTRODUCTORY  [ch. 

tion  among  living  things.  The  waves  of  the  sea,  the  little  ripples 
on  the  shore,  the  sweeping  curve  of  the  sandy  bay  between  the 
headlands,  the  outline  of  the  hills,  the  shape  of  the  clouds,  all  these 
are  so  many  riddles  of  form,  so  many  problems  of  morphology,  and 
all  of  them  the  physicist  can  more  or  less  easily  read  and  adequately 
solve:  solving  them  by  reference  to  their  antecedent  phenomena, 
in  the  material  system  of  mechanical  forces  to  which  they  belong, 
and  to  which  we  interpret  them  as  being  due.  They  have  also, 
doubtless,  their  immanent  teleological  significance;  but  it  is  on 
another  plane  of  thought  from  the  physicist's  that  we  contemplate 
their  intrinsic  harmony*  and  perfection,  and  "see  that  they  are 
good." 

Nor  is  it  otherwise  with  the  material  forms  of  living  things.  Cell 
and  tissue,  shell  and  bone,  leaf  and  flower,  are  so  many  portions  of 
matter,  and  it  is  in  obedience  to  the  laws  of  physics  that  their 
particles  have  been  moved,  moulded  and  conformedf.  They  are  no 
exception  to  the  rule  that  Qeos  aet  yeajjjLerpeL.  Their  problems  of 
form  are  in  the  first  instance  mathematical  problems,  their  problems 
of  growth  are  essentially  physical  problems,  and  the  morphologist  is, 
ipso  facto,  a  student  of  physical  science.  He  may  learn  from  that 
comprehensive  science,  as  the  physiologists  have  not  failed  to  do, 
the  point  of  view  from  which  her  problems  are  approached,  the 
quantitative  methods  by  which  they  are  attacked,  and  the  whole- 
some restraints  under  which  all  her  work  is  done.  He  may  come 
to  realise  that  there  is  no  branch  of  mathematics,  however  abstract, 
which  may  not  some  day  be  applied  to  phenomena  of  the  real 

*  What  I  understand  by  "holism"  is  what  the  Greeks  called  apfiovia.  This  is 
something  exhibited  not  only  by  a  lyre  in  tune,  but  by  all  the  handiwork  of 
craftsmen,  and  by  all  that  is  " put  together"  by  art  or  nature.  It  is  the  " composite- 
ness  of  any  composite  whole";  and,  like  the  cognate  terms  KpSicns  or  (rvvdeais,  implies 
a  balance  or  attunement.     Cf.  John  Tate,  in  Class.  Review,  Feb.  1939. 

t  This  general  principle  was  clearly  grasped  by  Mr  George  Rainey  many  years 
ago,  and  expressed  in  such  words  as  the  following:  "It  is  illogical  to  suppose  that 
in  the  case  of  vital  organisms  a  distinct  force  exists  to  produce  results  perfectly 
within  the  reach  of  physical  agencies,  especially  as  in  many  instances  no  end  could 
be  attained  were  that  the  case,  but  that  of  opposing  one  force  by  another  capable 
of  effecting  exactly  the  same  purpose."  (On  artificial  calculi,  Q.J. M.S.  {Trans. 
Microsc.  Soc),  vr,  p.  49,  18.58.)  Cf.  also  Helmholtz,  infra  cit.  p.  9.  (Mr  George 
Rainey,  a  man  of  learning  and  originality,  was  demonstrator  of  anatomy  at 
St  Thomas's;  he  followed  that  modest  calling  to  a  great  age,  and  is  remembered 
by  a  few  old  pupils  with  peculiar  affection.) 


I]  OF  EVOLUTION  AND  ENTROPY  11 

world*.  He  may  even  find  a  certain  analogy  between  the  slow, 
reluctant  extension  of  physical  laws  to  vital  phenomena  and  the  slow 
triumphant  demonstration  by  Tycho  Brahe,  Copernicus,  GaHleo  and 
Newton  (all  in  opposition  to  the  Aristotelian  cosmogony),  that  the 
heavens  are  formed  of  like  substance  with  the  earth,  and  that  the 
movements  of  both  are  subject  to  the  selfsame  laws. 

Organic  evolution  has  its  physical  analogue  in  the  universal  law 
that  the  world  tends,  in  all  its  parts  and  particles,  to  pass  from 
certain  less  probable  to  certain  more  probable  configurations  or 
states.  This  is  the  second  law  of  thermodynamics.  It  has  been 
called  the  law  of  evolution  of  the  world'f ;  and  we  call  it,  after  Clausius, 
the  Principle  of  Entropy,  which  is  a  literal  translation  of  Evolution 
into  Greek. 

The  introduction  of  mathematical  concepts  into  natural'  science 
has  seemed  to  many  men  no  mere  stumbling-block,  but  a  very 
parting  of  the  ways.  Bichat  was  a  man  of  genius,  who  did  immense 
service  to  philosophical  anatomy,  but,  like  Pascal,  he  utterly  refused 
to  bring  physics  or  mathematics  into  biology:  "  On  calcule  le  retour 
d'un  comete,  les  resistances  d'un  fluide  parcourant  un  canal  inerte, 
la  Vitesse  d'un  projectile,  etc.;  mais  calculer  avec  BorelU  la  force 
d'un  muscle,  avec  Keil  la  vitesse  du  sang,  avec  Jurine,  Lavoisier  et 
d'autres  la  quantite  d'air  entrant  dans  le  poumon,  c'est  batir  sur  un 
sable  mouvant  un  edifice  sohde  par  lui-meme,  mais  qui  tombe  bientot 
faute  de  base  assureej."  Comte  went  further  still,  and  said  that 
every  attempt  to  introduce  mathematics  into  chemistry  must  be 
deemed  profoundly  irrational,  and  contrary  to  the  whole  spirit  of 
the  science  §.  But  the  great  makers  of  modern  science  have  all  gone 
the  other  way.  Von  Baer,  using  a  bold  metaphor,  thought  that  it 
might  become  possible  "  die  bildenden  Krafte  des  thierischen  Korpers 
.  auf  die  allgemeinen  Krafte  oder  Lebenserscheinun^en  des  Weltganzes 
zuriickzufiihrenll."  Thomas  Young  shewed,  as  BorelU  had  done, 
how  physics  may  subserve  anatomy;  he  learned  from  the  heart  and 
a,rteries  that  "  the  mechanical  motions  which  take  place  in  an  animal's 
body  are  regulated  by  the  same  general  laws  as  the  motions  of 

*  So  said  Lobatchevsky. 

t  Cf.  Chwolson,  Lehrbuch,  iii,  p.  499,  1905;  J.  Perrin,  Traitd  de  chimie  physique, 
I,  p.  142,  1903;  and  Lotka's  Elements  of  Physical  Biology,  1925,  p.  26. 

t  La  Vie  et  la  Mort,  p.  81.  §  Philosophie  Positive,  Bk.  rv. 

]|    Ueber  Entwicklung  der  Thiere:  Beobachtungen  und  Reflexionen,  i,  p.  22,  1828. 


12  INTRODUCTORY  [ch. 

inanimate  bodies*."  And  Theodore  Schwann  said  plainly,  a  hun- 
dred years  ago,  "Ich  wiederhole  iibrigens  dass,  wenn  hier  von  einer 
physikahschen  Erklarung  der  organischen  Erscheinungen  die  Rede 
ist,  darunter  nicht  nothwendig  eine  Erklarung  durch  die  bekannten 
physikalischen  Krafte. .  .zu  verstehen  ist,  sondern  iiberhaupt  eine 
Erklarung  durch  Krafte,  die  nach  strengen  Gesetzen  der  blinden 
Nothwendigkeit  wie  die  physikalischen  Krafte  wirken,  mogen  diese 
Krafte  auch  in  der  anorganischen  Natur  auftreten  oder  nicht f." 

Helmholtz,  in  a  famous  and  influential  lecture,  and  surely  with 
these  very  words  of  Schwann's  in  mind,  laid  it  down  as  the  funda- 
mental principle  of  physiology  that  "there  may  be  other  agents 
acting  in  the  hving  body  than  those  agents  which  act  in  the  inorganic 
world ;  but  these  forces,  so  far  as  they  cause  chemical  and  mechanical 
influence  in  the  body,  must  be  quite  of  the  same  character  as  inorganic 
forces :  in  this,  at  least,  that  their  eff'ects  must  be  ruled  by  necessity, 
and  must  always  be  the  same  when  acting  under  the  same  conditions ; 
and  so  there  cannot  exist  any  arbitrary  choice  in  the  direction  of  their 
actions."  It  follows  further  that,  like  the  other  "physical"  forces, 
they  must  be  subject  to  mathematical  analysis  and  deduction  J. 

So  much  for  the  physico-chemical  problems  of  physiology.  Apart 
from  these,  the  road  of  physico-mathematical  or  dynamical  investi- 
gation in  morphology  has  found  few  to  follow  it;  but  the  pathway 
is  old.  The  way  of  the  old  Ionian  physicians,  of  Anaxagoras  § ,  of 
Empedocles  and  his  disciples  in  the  days  before  Aristotle,  lay  just 
by  that  highway  side.  It  was  Galileo's  and  Borelli's  way;  and 
Harvey's  way,  when  he  discovered  the  circulation  of  the  blood ||. 
It  was  little  trodden  for  long  afterwards,  but  once  in  a  while 
Swammerdam  and  Reaumur  passed  thereby.  And  of  later  years 
Moseley  and  Meyer,  Berthold,  Errera  and  Roux  have  been  among 

*  Croonian  Lecture  on  the  heart  and  arteries,  Phil.  Trans.  1809,  p.  1;  Collected 
Works,  I,  p.  511. 

t  M ikroskopische  Untersuchungen,  1839,  p.  226. 

J  The  conservation  of  forces  applied  to  organic  nature,  Proc.  Royal  Inst. 
April  12,  1861. 

§  Whereby  he  incurred  the  reproach  of  Socrates,  in  the  Phaedo.  See  Clerk 
Maxwell  on  Anaxagoras  as  a  Physicist,  in  Phil.  Mag.  (4),  xlvi,  pp.  453-460,  1873. 

II  Cf.  Harvey's  preface  to  his  Exercitationes  de  Generatione  Animalium,  1651: 
"Quoniam  igitur  in  Generatione  animalium  (ut  etiam  in  caeteris  rebus  omnibus 
de  quibus  aliquid  scire  cupimus),  inquisitio  omnis  a  caussis  petenda  est,  praesertim 
a  materiali  et  efficiente:   visum  est  mihi"  etc. 


I]  OF  NATURAL  PHILOSOPHY  13 

the  little  band  of  travellers.  We  need  not  wonder  if  the  way  be 
hard  to  follow,  and  if  these  wayfarers  have  yet  gathered  little. 
A  harvest  has  been  reaped  by  others,  and  the  gleaning  of  the  grapes 
is  slow. 

It  behoves  us  always  to  remember  that  in  physics  it  has  taken 
great  men  to  discover  simple  things.  They  are  very  great  names 
indeed  which  we  couple  with  the  explanation  of  the  path  of  a  stone, 
the  droop  of  a  chain,  the  tints  of  a  bubble,  the  shadows  in  a  cup. 
It  is  but  the  shghtest  adumbration  of  a  dynamical  morphology  that 
we  can  hope  to  have  until  the  physicist  and  the  mathematician  shall 
have  made  these  problems  of  ours  their  own,  or  till  a  new  Boscovich  shall 
have  written  for  the  naturahst  the  new  TheariaPhilosophiaeNaturalis. 

How  far  even  then  mathematics  will  suffice  to  describe,  and 
physics  to  explain,  the  fabric  of  the  body,  no  man  can  foresee.  It 
may  be  that  all  the  laws  of  energy,  and  all  the  properties  of  matter, 
and  all  the  chemistry  of  all  the  colloids  are  as  powerless  to  explain 
the  body  as  they  are  impotent  to  comprehend  the  soul.  For  my 
part,  I  think  it  is  not  so.  Of  how  it  is  that  the  soul  informs  the 
body,  physical  science  teaches  me  nothing;  and  that  living  matter 
influences  and  is  influenced  by  mind  is  a  mystery  without  a  clu'e. 
Consciousness  is  not  explained  to  my  comprehension  by  all  the 
nerve-paths  and  neurones  of  the  physiologist ;  nor  do  I  ask  of  physics 
how  goodness  shines  in  one  man's  face,  and  evil  betrays  itself  in 
another.  But  of  the  construction  and  growth  and  working  of  the 
body,  as  of  all  else  that  is  of  the  earth  earthy,  physical  science  is, 
in  my  humble  opinion,  our  only  teacher  and  guide. 

Often  and  often  it  happens  that  our  physical  knowledge  is  in- 
adequate to  explain  the  mechanical  working  of  the  organism;  the 
phenomena  are  superlatively .  complex,  the  procedure  is  involved 
and  entangled,  and  the  investigation  has  occupied  but  a  few  short 
lives  of  men.  When  physical  science  falls  short  of  explaining  the 
order  which  reigns  throughout  these  manifold  phenomena — an  order 
more  characteristic  in  its  totality  than  any  of  its  phenomena  in 
themselves — men  hasten:  to  invoke  a  guiding  principle,  an  entelechy, 
or  call  it  what  you  will.  But  all  the  while  no  physical  law,  any 
more  than  gravity  itself,  not  even  among  the  puzzles  of  stereo- 
chemistry or  of  physiological  surface-action  and  osmosis,  is  known 
to  be  transgressed  by  the  bodily  mechanism. 


14  INTRODUCTORY  [ch. 

Some  physicists  declare,  as  Maxwell  did,  that  atoms  or  molecules 
more  compHcated  by  far  than  the  chemist's  hypotheses  demand,  are 
requisite  to  explain  the  phenomena  of  life.  If  what  is  impHed  be 
an  explanation  of  psychical  phenomena,  let  the  point  be  granted  at 
once;  we  may  go  yet  further  and  decHne,  with  Maxwell,  to  believe 
that  anything  of  the  nature  of  physical  complexity,  however  exalted, 
could  ever  suffice.  Other  physicists,  like  Auerbach*,  or  Larmorj, 
or  Joly  J,  assure  us  that  our  laws  of  thermodynamics  do  not  suffice, 
or  are  inappropriate,  to  explain  the  maintenance,  or  (in  Joly's  phrase) 
the  accelerative  absorption,  of  the  bodily  energies,  the  retardation 
of  entropy,  and  the  long  battle  against  the  cold  and  darkness  which 
is  death.  With  these  weighty  problems  I  am  not  for  the  moment 
concerned.  My  sole  purpose  is  to  correlate  with  mathematical  state- 
ment and  physical  law  certain  of  the  simpler  outward  phenomena 
of  organic  growth  and  structure  or  form,  while  all  the  while  regarding 
the  fabric  of  the  organism,  ex  hypothesi,  as  a  material  and  mechanical 
configuration.  This  is  my  purpose  here.  But  I  would  not  for  the 
world  be  thought  to  beheve  that  this  is  the  only  story  which  Life 
and  her  Children  have  tp  tell.  One  does  not  come  by  studying 
living  things  for  a  lifetime  to  suppose  that  physics  and  chemistry 
can  account  for  them  all§. 

Physical  science  and  philosophy  stand  side  by  side,  and  one 
upholds  the  other.  Without  something  of  the  strength  of  physics 
philosophy  would  be  weak ;  and  without  something  of  philosophy's 
wealth  physical  science  would  be  poor.  "Rien  ne  retirera  du  tissu 
de  la  science  les  fils  d'or  que  la  main  du  philosophe  y  a  introduits||." 
But  there  are  fields  where  each,  for  a  while  at  least,  must  work  alone; 
and  where  physical  science  reaches  its  limitations  physical  science 
itself  must  help  us  to  discover.     Meanwhile  the  appropriate  and 

*  Ektropismus,  oder  die  physikalische  Theorie  des  Lebens,  Leipzig,  1810. 

t  Wilde  Lecture,  Nature,  March  12,  1908;  ibid.  Sept.  6,  1900;  Aether  and  Matter, 
p.  288.     Cf.  also  Kelvin,  Fortnightly  Review,  1892,  p.  313. 

X  The  abundance  of  life,  Proc.  Roy.  Dublin  Soc.  vii,  1890;  Scientific  Essays, 
1915,  p.  60  seq. 

§  That  mechanism  has  its  share  in  the  scheme  of  nature  no  philosopher  has 
denied.  Aristotle  (or  whosoever  wrote  the  De  Mundo)  goes  so  far  as  to  assert  that 
in  the  most  mechanical  operations  of  nature  we  behold  some  of  the  divinest 
attributes  of  God. 

II  J.  H,  Fr.  Papillon,  Histoire  de  la  jyhilosophie  moderne  dans  ses  rapports  avec  le 
developpement  des  sciences  de  la  nature,  i,  p.  300,  1870. 


I]  OF  LIFE  ITSELF  15 

legitimate  postulate  of  the  physicist,  in  approaching  the  physical 
problems  of  the  living  body,  is  that  with  these  physical  phenomena 
no  ahen  influence  interferes.  But  the  postulate,  though  it  is  certainly 
legitimate,  and  though  it  is  the  proper  and  necessary  prelude  to 
scientific  enquiry,  may  some  day  be  proven  to  be  untrue;  and  its 
disproof  will  not  be  to  the  physicist's  confusion,  but  will  come  as 
his  reward.  In  dealing  with  forms  which  are  so  concomitant  with 
life  that  they  are  seemingly  controlled  by  life,  it  is  in  no  spirit  of 
arrogant  assertiveness  if  the  physicist  begins  his  argument,  after  the 
fashion  of  a  most  illustrious  exemplar,  with  the  old  formula  of 
scholastic  challenge:   An  Vita  sit?     Dico  quod  non. 

The  terms  Growth  and  Form,  which  make  up  the  title  of  this  book, 
are  to  be  understood,  as  I  need  hardly  say,  in  their  relation  to  the 
study  of  organisms.  We  want  to  see  how,  in  some  cases  at  least, 
the  forms  of  living  things,  and  of  the  parts  of  living  things,  can  be 
explained  by  physical  considerations,  and  to  realise  that  in  general 
no  organic  forms  exist  save  such  as  are  in  conformity  with  physical 
and  mathematical  laws.  And  while  growth  is  a  somewhat  vague 
word  for  a  very  complex  matter,  which  may  depend  on  various 
things,  from  simple  imbibition  of  water  to  the  complicated  results 
of  the  chemistry  of  nutrition,  it  deserves  to  be  studied  in  relation 
to  form :  whether  it  proceed  by  simple  increase  of  size  without  obvious 
alteration  of  form,  or  whether  it  so  proceed  as  to  bring  about  a 
gradual  change  of  form  and  the  slow  development  of  a  more  or  less 
complicated  structure. 

In  the  Newtonian  language*  of  elementary  physics,  force  is 
recognised  by  its  action  in  producing  or  in  changing  motion,  or 
in  preventing  change  of  motion  or  in  maintaining  rest.  When  we 
deal  with  matter  in  the  concrete,  force  does  not,  strictly  speaking, 
enter  into  the  question,  for  force,  unlike  matter,  has  no  independent 
objective  existence.  It  is  energy  in  its  various  forms,  known  or 
unknown,  that  acts  upon  matter.  But  when  we  abstract  our 
thoughts  from  the  material  to  its  form,  or  from  the  thing  moved  to 
its  motions,  when  we  deal  with  the  subjective  conceptions  of  form, 

*  It  is  neither  unnecessary  nor  superfluous  to  explain  that  physics  is  passing 
through  an  empirical  phase  into  a  phase  of  pure  mathematical  reasoning.  But 
when  we  use  physics  to  interpret  and  elucidate  our  biology,  it  is  the  old-fashioned 
empirical  physics  which  we  endeavour,  and  are  alone  able,  to  apply. 


16  INTRODUCTORY  [ch. 

or  movement,  or  the  movements  that  change  of  form  impHes,  then 
Force  is  the  appropriate  term  for  our  conception  of  the  causes  by 
which  these  forms  and  changes  of  form  are  brought  about.  When 
we  use  the  term  force,  we  use  it,  as  the  physicist  always  does,  for 
the  sake  of  brevity,  using  a  symbol  for  the  magnitude  and  direction 
of  an  action  in  reference  to  the  symbol  or  diagram  of  a  material 
thing.  It  is  a  term  as  subjective  and  symbolic  as  form  itself,  and 
SO'  is  used  appropriately  in  connection  therewith. 

The  form,  then,  of  any  portion  of  matter,  whether  it  be  living 
or  dead,  and  the  changes  of  form  which  are  apparent  in  its  movements 
and  in  its  growth,  may  in  all  cases  alike  be  described  as  due  to 
the  action  of  force.  In  short,  the  form  of  an  object  is  a  "diagram 
of  forces,"  in  this  sense,  at  least,  that  from  it  we  can  judge  of  or 
deduce  the  forces  that  are  acting  or  have  acted  upon  it:  in  this 
strict  and  particular  sense,  it  is  a  diagram — in  the  case  of  a  sohd, 
of  the  forces  which  have  been  impressed  upon  it  when  its  conformation 
was  produced,  together  with  those  which  enable  it  to  retain  its 
conformation;  in  the  case  of  a  Hquid  (or  of  a  gas)  of  the  forces  which 
are  for  the  moment  acting  on  it  to  restrain  or  balance  its  own 
inherent  mobility.  In  an  organism,  great  or  small,  it  is  not  merely 
the  nature  of  the  motions  of  the  hving  substance  which  we  must 
interpret  in  terms  of  force  (according  to  kinetics),  but  also  the 
conformation  of  the  organism  itself,  whose  permanence  or  equilibrium 
is  explained  by  the  interaction  or  balance  of  forces,  as  described  in 
statics. 

If  we  look  at  the  hving  cell  of  an  Amoeba  or  a  Spirogyra,  we 
see  a  something  which  exhibits  certain  active  movements,  and  a 
certain  fluctuating,  or  more  or  less  lasting,  form;  and  its  form  at 
a  given  moment,  just  like  its  motions,  is  to  be  investigated  by  the 
help  of  physical  methods,  and  explained  by  the  invocation  of  the 
mathematical  conception  of  force. 

Now  the  state,  including  the  shape  or  form,  of  a  portion  of  matter 
is  the  resultant  of  a  number  of  forces,  which  represent  or  symbolise 
the  manifestations  of  various  kinds  of  energy;  and  it  is  obvious, 
accordingly,  that  a  great  part  of  physical  science  must  be  under- 
stood or  taken  for  granted  as  the  necessary  preliminary  to  the 
discussion  on  which  we  are  engaged.  But  we  may  at  least  try  to 
indicate,  very  briefly,  the  nature  of  the  principal  forces  and  the 


I]  OF  MATTER  AND  ENERGY  17 

principal  properties  of  matter  with  which  our  subject  obhges  us  to 
deal.  Let  us  imagine,  for  instance,  the  case  of  a  so-called  "simple" 
organism,  such  as  Amoeba;  and  if  our  short  list  of  its  physical 
•properties  and  conditions  be  helpful  to  our  further  discussion,  we 
need  not  consider  how  far  it  be  complete  or  adequate  from  the 
wider  physical  point  of  view*. 

This  portion  of  matter,  then,  is  kept  together  by  the  inter- 
molecular  force  of  cohesion;  in  the  movements  of  its  particles 
relatively  to  one  another,  and  in  its  own  movements  relative  to 
adjacent  matter,  it  meets  with  the  opposing  force  of  friction — 
without  the  help  of  which  its  creeping  movements  could  not  be 
performed.  It  is  acted  on  by  gravity,  and  this  force  tends  (though 
slightly,  owing  to  the  Amoeba's  small  mass,  and  to  the  small 
difference  between  its  density  and  that  of  the  surrounding  fluid) 
to  flatten  it  down  upon  the  solid  substance  on  which  it  may  be 
creeping.  Our  Amoeba  tends,  in  the  next  place,  to  be  deformed 
by  any  pressure  from  outside,  even  though  slight,  which  may  be 
applied  to  it,  and  this  circumstance  shews  it  to  consist  of  matter 
in  a  fluid,  or  at  least  semi-fluid,  state:  which  state  is  further 
indicated  when  we  observe  streaming  or  current  motions  in  its 
interior.  Like  other  fluid  bodies,  its  surfacef,  whatsoever  other 
substance — gas,  hquid  or  solid — it  be  in  contact  with,  and  in  varying 
degree  according  to  the  nature  of  that  adjacent  substance,  is  the 
seat  of  molecular  force  exhibiting  itself  as  a  surface-tension,  from 
the  action  of  which  many  important  consequences  follow,  greatly 
affecting  the  form  of  the  fluid  surface. 

While  the  protoplasmj  of  the  Amoeba  reacts  to  the  shghtest 
pressure,  and  tends  to  "flow,"  and  while  we  therefore  speak  of  it 

*  With  the  special  and  impprtant  properties  of  colloidal  matter  we  are,  for 
the  time  being,  not  concerned. 

t  Whether  an  animal  cell  has  a  membrane,  or  only  a  pellicle  or  zona  limitans, 
was  once  deemed  of  great  importance,  and  played  a  big  part  in  the  early  contro- 
versies between  the  cell-theory  of  Schwann  and  the  protoplasma-theory  of  Max 
Schultze  and  others,  Dujardin  came  near  the  truth  when  he  said,  somewhat 
naively,  "en  niant  la  presence  d'un  tegument  propre,  je  ne  pretends  pas  du  tout 
nier  i'existence  d'une  surface." 

%  The  word  protoplasm  is  used  here  in  its  most  general  sense,  as  vaguely  as  when 
Huxley  spoke  of  it  as  the  "physical  basis  of  life."  Its  many  changes  and  shades 
of  meaning  in  early  years  are  discussed  by  Van  Bambeke  in  the  Bull.  Sac.  Beige 
de  Microscopie,  xxn,  pp.  1-16,  1896. 


18  INTRODUCTORY  [ch. 

as  a  fluid*,  it  is  evidently  far  less  mobile  than  such  a  fluid  (for 
instance)  as  water,  but  is  rather  Uke  treacle  in  its  slow  creeping 
movements  as  it  changes  its  shape  in  response  to  force.  Such  fluids 
are  said  to  have  a  high  viscosity,  and  this  viscosity  obviously  acts 
in  the  way  of  resisting  change  of  form,  or  in  other  words  of 
retarding  the  efl'ects  of  any  disturbing  action  of  force.  When  the 
viscous  fluid  is  capable  of  being  drawn  out  into  fine  threads,  a 
property  in  which  we  know  that  some  Amoebae  differ  greatly  from 
others,  we  say  that  the  fluid  is  also  viscid,  or  exhibits  viscidity. 
Again,  not  by  virtue  of  our  Amoeba  being  liquid,  but  at  the  same 
time  in  vastly  greater  measure  than  if  it  were  a  sohd  (though  far  less 
rapidly  than  if  it  were  a  gas),  a  process  of  molecular  diffusion  is 
constantly  going  on  within  its  substance,  by  which  its  particles 
interchange  their  places  within  the  mass,  while  surrounding  fluids, 
gases  and  soUds  in  solution  diffuse  into  and  out  of  it.  In  so  far 
as  the  outer  wall  of  the  cell  is  different  in  character  from  the 
interior,  whether  it  be  a  mere  pelhcle  as  in  Amoeba  or  a  firm 
cell-wall  as  in  Protococcus,  the  diffusion  which  takes  place  throtigh 
this  wall  is  sometimes  distinguished  under  the  term  osmosis. 

Within  the  cell,  chemical  forces  are  at  work,  and  so  also  in  all 
probabihty  (to  judge  by  analogy)  are  electrical  forces;  and  the 
organism  reacts  also  to  forces  from  without,  that  have  their  origin 
in  chemical,  electrical  and  thermal  influences.  The  processes  of 
diffusion  and  of  chemical  activity  within  the  cell  result,  by  the 
drawing  in  of  water,  salts,  and  food-material  with  or  without 
chemical  transformation  into  protoplasm,  in  growth,  and  this  com- 
plex phenomenon  we  shall  usually,  without  discussing  its  nature 
and  origin,  describe  and  picture  as  a  force.  Indeed  we  shall 
manifestly  be  incHned  to  use  the  term  growth  in  two  senses,  just 
indeed  as  we  do  in  the  case  of  attraction  or  gravitation,  on  the  one 
hand  as  a  process,  and  on  the  other  as  a  force. 

In  the  phenomena  of  cell-division,  in  the  attractions  or  repulsions 
of  the  parts  of  the  dividing  nucleus,  and  in  the  "  caryokinetic " 
figures  which  appear  in  connection  with  it,  we  seem  to  see  in 
operation  forces  and  the  effects  of  forces  which  have,  to  say  the 

*  One  of  the  first  statements  which  Dujardin  made  about  protoplasm  (or,  as 
he  called  it,  sarcode)  was  that  it  was  not  a  fluid;  and  he  relied  greatly  on  this  fact 
to  shew  that  it  was  a  living,  or  an  organised,  structure. 


I]  OF  VITAL  PHENOMENA  19 

least  of  it,  a  close  analogy  with  known  physical  phenomena :  and 
to  this  matter  we  shall  presently  return.  But  though  they  resemble 
known  physical  phenomena,  their  nature  is  still  the  subject  of  much 
dubiety  and  discussion,  and  neither  the  forms  produced  nor  the 
forces  at  work  can  yet  be  satisfactorily  and  simply  explained.  We 
may  readily  admit  then,  that,  besides  phenomena  which  are  obviously 
physical  in  their  nature,  there  are  actions  visible  as  well  as  invisible 
taking  place  within  living  cells  which  our  knowledge  does  not  permit 
us  to  ascribe  with  certainty  to  any  known  physical  force;  and  it 
may  or  may  not  be  that  these  phenomena  will  yield  in  time  to  the 
methods  of  physical  investigation.  Whether  they  do  or  no,  it  is 
plain  that  we  have  no  clear  rule  or  guidance  as  to  what  is  "vital" 
and  what  is  not;  the  whole  assemblage  of  so-called  vital  phenomena, 
or  properties  of  the  organism,  cannot  be  clearly  classified  into  those 
that  are  physical  in  origin  and  those  that  are  sui  generis  and  peculiar 
to  living  things.  All  we  can  do  meanwhile  is  to  analyse,  bit  by  bit, 
those  parts  of  the  whole  to  which  the  ordinary  laws  of  the  physical 
forces  more  or  less  obviously  and  clearly  and  indubitably  apply. 

But  even  the  ordinary  laws  of  the  physical  forces  are  by  no  means 
simple  and  plain.  In  the  winding  up  of  a  clock  (so  Kelvin  once 
said),  and  in  the  properties  of  matter  which  it  involves,  there  is 
enough  and  more  than  enough  of  mystery  for  our  limited  under- 
standing: "a  watchspring  is  much  farther  beyond  our  understanding 
than  a  gaseous  nebula."  We  learn  and  learn,  but  never  know  all, 
about  the  smallest,  humblest  thing.  So  said  St  Bonaventure :  "  Si  per 
multos  annos  viveres,  adhuc  naturam  unius  festucae  seu  muscae  seu 
minimae  creaturae  de  mundo  ad  plenum  cognoscere  non  valeres*." 
There  is  a  certain  fascination  in  such  ignorance ;  and  we  learn  (like 
the  Abbe  Galiani)  without  discouragement  that  Science  is  "plutot 
destine  a  etudier  qu'a  connaitre,  a  chercher  qu'a  trouver  la  verite." 

Morphology  is  not  only  a  study  of  material  things  and  of  the  forms 
of  material  things,  but  has  its  dynamical  aspect,  under  which  we 
deal  with  the  interpretation,  in  term^  of  force,  of  the  operations  of 
Energyf .     And  here  it  is  well  worth  while  to  remark  that,  in  deahng 

*  Op.  V,  p.  541 ;  cit.  E.  Gilson. 

t  This  is  a  great  theme.  Boltzmann,  writing  in  1886  on  the  second  law  of 
thermodynamics,  declared  that  available  energy  was  the  main  object  at  stake 
in  the  struggle  for  existence  and  the  evolution  of  the  world.  Cf.  Lotka,  The 
energetics  of  evolution,  Proc.  Nat.  Acad.  Sci.  1922,  p.  147. 


20  INTRODUCTORY  [ch. 

with  the  facts  of  embryology  or  the  phenomena  of  inheritance,  the 
common  language  of  the,  books  seems  to  deal  too  much  with  the 
material  elements  concerned,  as  the  causes  of  development,  of 
variation  or  of  hereditary  transmission.  Matter  as  such  produces 
nothing,  changes  nothing,  does  nothing;  and  however  convenient 
it  may  afterwards  be  to  abbreviate  our  nomenclature  and  our 
descriptions,  we  must  most  carefully  realise  in  the  outset  that  the 
spermatozoon,  the  nucleus,  the  chromosomes  or  the  germ-plasma 
can  never  act  as  matter  alone,  but  only  as  seats  of  energy  and  as 
centres  of  force.  And  this  is  but  an  adaptation  (in  the  light,  or 
rather  in  the  conventional  symboHsm,  of  modern  science)  of  the  old 
saying  of  the  philosopher :  apx^  yo.p  r)  <j>voLs  fxdXXov  rrjs  vXrjg. 

Since  this  book  was  written,  some  five  and  twenty  years  ago, 
certain  great  physico-mathematical  concepts  have  greatly  changed. 
Newtonian  mechanics  and  Newtonian  concepts  of  space  and  time 
are  found  unsuitable,  even  untenable  or  invahd,  for  the  all  but 
infinitely  great  and  the  all  but  infinitely  small.  The  very  idea  of 
physical  causation  is  said  to  be  illusory,  and  the  physics  of  the 
atom  and  the  electron,  and  of  the  quantum  theory,  are  to  be 
elucidated  by  the  laws  of  probability  rather  than  by  the  concept 
of  causation  and  its  effects.  But  the  orders  of  magnitude,  whether 
of  space  or  time,  within  which  these  new  concepts  become  useful, 
or  hold  true,  lie  far  away.  We  distinguish,  and  can  never  help 
distinguishing,  between  the  things  which  are  of  our  own  scale  and 
order,  to  which  our  minds  are  accustomed  and  our  senses  attuned, 
and  those  remote  phenomena  which  ordinary  standards  fail  to 
measure,  in  regions  where  (as  Robert  Louis  Stevenson  said)  there 
is  no  habitable  city  for  the  mind  of  man. 

It  is  no  wonder  if  new  methods,  new  laws,  new  words,  new  modes 
of  thought  are  needed  when  we  make  bold  to  contemplate  a  Universe 
within  which  all  Newton's  is  but  a  speck.  But  the  world  of  the 
Hving,  wide  as  it  may  be,  is  bounded  by  a  famihar  horizon  within 
which  our  thoughts  and  senses  are  at  home,  our  scales  of  time  and 
magnitude  suffice,  and  the  Natural  Philosophy  of  Newton  and 
Gahleo  rests  secure. 

We  start,  like  Aristotle,  with  our  own  stock-in-trade  of  know- 
ledge: dpKTeov  OLTTO  Tcov  rjfjuv  yvajpLjjLojv.     And  only  when  we  are 


I]  OF  NEWTONIAN  PHYSICS  21 

steeped  to  the  marrow  (as  Henri  Poincare  once  said)  in  the  old  laws, 
and  in  no  danger  of  forgetting  them,  may  we  be  allowed  to  learn 
how  they  have  their  remote  but  subtle  limitations,  and  cease  afar 
off  to  be  more  than  approximately  true  *.  Kant's  axiom  of  causahty, 
that  it  is  denknotwendig — indispensable  for  thought — remains  true 
however  physical  science  may  change.  His  later  aphorism,  that  all 
changes  take  place  subject  to  the  law  which  links  cause  and  effect 
together — "alle  Veranderungen  geschehen  nach  dem  Gesetz  der 
Verkniipfung  von  Ursache  und  Wirkung" — is  still  an  axiom  a  priori, 
independent  of  experience:  for  experience  itself  depends  upon  its 
truth  t- 

*  So  Max  Planck  himself  says  somewhere:  "In  my  opinion  the  teaching  of 
mechanics  will  still  have  to  begin  with  Newtonian  force,  just  as  optics  begins  in 
the  sensation  of  colour  and  thermodynamics  with  the  sensation  of  warmth, 
despite  th^  fact  that  a  more  precise  basis  is  substituted  later  on." 

t  "Weil  er  [der  Grundsatz  das  Kausalverhaltnisses]  selbst  der  grund  der  Moglich- 
keit  einer  solchen  Erfahrung  ist":  Kritik  d.  reinen  Vernunft,  ed.  Odicke,  1889,  p.  221. 
Cf.  also  G.  W.  Kellner,  Die  Kausalitat  in  der  Physik,  Ztschr.f.  Physik,  lx,iv,  pp.  568- 
580.  1930. 


CHAPTER  II 

ON  MAGNITUDE 

To  terms  of  magnitude,  and  of  direction,  must  we  refer  all  our 
conceptions  of  Form.  For  the  form  of  an  object  is  defined  when  we 
know  its  magnitude,  actual  or  relative,  in  various  directions;  and 
Growth  involves  the  same  concepts  of  magnitude  and  direction, 
related  to  the  further  concept,  or  "dimension,"  of  Time.  Before 
we  proceed  to  the  consideration  of  specific  form,  it  will  be  well  to 
consider  certain  general  phenomena  of  spatial  magnitude,  or  of  the 
extension  of  a  body  in  the  several  dimensions  of  space. 

We  are  taught  by  elementary  mathematics — and  by  Archimedes 
himself — that  in  similar  figures  the  surface  increases  as  the  square, 
and  the  volume  as  the  cube,  of  the  linear  dimensions.  If  we  take 
the  simple  case  of  a  sphere,  with  radius  r,  the  area  of  its  surface  is 
equal  to  4:7rr^,  and  its  volume  to  ^ttt^  ;  from  which  it  follows  that  the 
ratio  of  its  volume  to  surface,  or  V/S,  is  Jr.  That  is  to  say,  VfS 
varies  as  r;  or,  in  other  words,  the  larger  the  sphere  by  so  much  the 
greater  will  be  its  volume  (or  its  mass,  if  it  be  uniformly  dense 
throughout)  in  comparison  with  its  superficial  area.  And,  taking 
L  to  represent  any  linear  dimension,  we  may  write  the  general 
equations  in  the  form 

Soz  L\  F  oc  L^ 

or  iS  =  kL\  and   V  =  k'L\ 

where  k,  k',  are  "factors  of  proportion," 

V  V       k 

and  ^  cc  L,      or      —  =  j-,  L  =  KL. 

o  ok 

So,  in  Lilliput,  "His  Majesty's  Ministers,  finding  that  Gulhver's 
stature  exceeded  theirs  in  the  proportion  of  twelve  to  one,  concluded 
from  the  similarity  of  their  bodies  that  his  must  contain  at  least 
1728  [or  12^]  of  theirs,  and  must  needs  be  rationed  accordingly*." 

*  Likewise  Gulliver  had  a  whole  Lilliputian  hogshead  for  his  half-pint  of  wine: 
in  the  due  proportion  of  1728  half-pints,  or  108  gallons,  equal  to  one  pipe  or 


CH.  II]  OF  DIMENSIONS  23 

From  these  elementary  principles  a  great  many  consequences 
follow,  all  more  or  less  interesting,  and  some  of  them  of  great 
importance.  In  the  first  place,  though  growth  in  length  (let  us  say) 
and  growth  in  volume  (which  is  usually  tantamount  to  mass  or 
weight)  are  parts  of  one  and  the  same  process  or  phenomenon,  the 
one  attracts  our  attention  by  its  increase  very  much  more  than  the 
other.  For  instance  a  fish,  in  doubhng  its  length,  multiphes  its 
weight  no  less  than  eight  times;  and  it  all  but  doubles  its  weight  in 
growing  from  four  inches  long  to  five. 

In  the  second  place,  we  see  that  an  understanding  of  the  correla- 
tion between  length  and  weight  in  any  particular  species  of  animal, 
in  other  words  a  determination  of  k  in  the  formula  W  =  k.L^, 
enables  us  at  any  time  to  translate  the  one  magnitude  into  the  other, 
and  (so  to  speak)  to  weigh  the  animal  with  a  measuring-rod;  this, 
however,  being  always  subject  to  the  condition  that  the  animal  shall 
in  no  way  have  altered  its  form,  nor  its  specific  gravity.  That  its 
specific  gravity  or  density  should  materially  or  rapidly  alter  is  not 
very  likely;  but  as  long  as  growth  lasts  changes  of  form,  even 
though  inappreciable  to  the  eye,  are  apt  and  hkely  to  occur.  Now 
weighing  is  a  far  easier  and  far  more  accurate  operation  than 
measuring;  and  the  measurements  which  would  reveal  slight  and 
otherwise  imperceptible  changes  in  the  form  of  a  fish — slight  relative 
differences  between  length,  breadth  and  depth,  for  instance — would 
need  to  be  very  dehcate  indeed.  But  if  we  can  make  fairly  accurate 
determinations  of  the  length,  which  is  much  the  easiest  linear 
dimension  to  measure,  and  correlate  it  with  the  weight,  then  the 
value  of  k,  whether  it  varies  or  remains  constant,  will  tell  us  at  once 
whether  there  has  or  has  not  been  a  tendency  to  alteration  in  the 
general  form,  or,  in  other  words,  a  difference  in  the  rates  of  growth 
in  different  directions.  To  this  subject  we  shall  return,  when  we 
come  to  consider  more  particularly  the  phenomenon  of  rate  of  growth. 

double-hogshead.  But  Gilbert  White  of  Selborne  could  not  see  what  was  plain 
to  the  Lilliputians;  for  finding  that  a  certain  little  long-legged  bird,  the  stilt, 
weighed  4J  oz.  and  had  legs  8  in.  long,  he  thought  that  a  flamingo,  weighing  4  lbs., 
should  have  legs  10  ft.  long,  to  be  in  the  same  proportion  as  the  stilt's.  But 
it  is  obvious  to  us  that,  as  the  weights  of  the  two  birds  are  as  1  :  15,  so  the  legs 
(or  other  linear  dimensions)  should  be  as  the  cube-roots  of  these  numbers,  or 
nearly  as  1  :  2^.  And  on  this  scale  the  flamingo's  legs  should  be,  as  they  actually 
are,  about  20  in.  long. 


24  ON  MAGNITUDE  [ch. 

We  are  accustomed  to  think  of  magnitude  as  a  purely  relative 
matter.  We  call  a  thing  big  or  little  with  reference  to  what  it  is 
wont  to  be,  as  when  we  speak  of  a  small  elephant  or  a  large  rat ;  and 
we  are  apt  accordingly  to  suppose  that  size  makes  no  other  or  more 
essential  difference,  and  that  Lilliput  and  Brobdingnag*  are  all 
alike,  according  as  we  look  at  them  through  one  end  of  the  glass 
or  the  other.  Gulliver  himself  declared,  in  Brobdingnag,  that 
"undoubtedly  philosophers  are  in  the  right  when  they  tell  us  that 
nothing  is  great  and  little  otherwise  than  by  comparison":  and 
Oliver  Heaviside  used  to  say,  in  like  manner,  that  there  is  no 
absolute  scale  of  size  in  the  Universe,  for  it  is  boundless  towards 
the  great  and  also  boundless  towards  the  small.  It  is  of  the  very 
essence  of  the  Newtonian  philosophy  that  we  should  be  able  to 
extend  our  concepts  and  deductions  from  the  one  extreme  of  magni- 
tude to  the  other;  and  Sir  John  Herschel  said  that  "the  student 
must  lay  his  account  to  finding  the  distinction  of  great  and  little 
altogether  annihilated  in  nature." 

All  this  is  true  of  number,  and  of  relative  magnitude.  The  Universe 
has  its  endless  gamut  of  great  and  small,  of  near  and  far,  of  many 
and  few.  Nevertheless,  in  physical  science  the  scale  of  absolute 
magnitude  becomes  a  very  real  and  important  thing;  and  a  new 
and  deeper  interest  arises  out  of  the  changing  ratio  of  dimensions 
when  we  come  to  consider  the  inevitable  changes  of  physical  rela- 
tions with  which  it  is  bound  up.  The  effect  of  scale  depends  not  on 
a  thing  in  itself,  but  in  relation  to  its  whole  environment  or  milieu ; 
it  is  in  conformity  with  the  thing's  "place  in  Nature,"  its  field  of 
action  and  reaction  in  the  Universe.  Everywhere  Nature  works 
true  to  scale,  and  everything  has  its  proper  size  accordingly.  Men 
and  trees,  birds  and  fishes,  stars  and  star-systems,  have  their 
appropriate  dimensions,  and  their  more  or  less  narrow  range  of 
absolute  magnitudes.  The  scale  of  human  observation  and  ex- 
perience lies  within  the  narrow  bounds  of  inches,  feet  or  miles,  all 
measured  in  terms  drawn  from  our  own  selves  or  our  own  doings. 
Scales  which  include  light-years,  parsecs.  Angstrom  units,  or  atomic 

*  Swift  paid  close  attention  to  the  arithmetic  of  magnitude,  but  none  to  its 
physical  aspect.  See  De  Morgan,  on  Lilliput,  in  N.  and  Q.  (2),  vi,  pp.  123-125, 
1858.  On  relative  magnitude  see  also  Berkeley,  in  his  Essay  towards  a  New  Theory 
of  Visio7i,  1709. 


II]  THE  EFFECT  OF  SCALE  25 

and  sub-atomic  magnitudes,  belong  to  other  orders  of  things  and 
other  principles  of  cognition. 

A  common  effect  of  scale  is  due  to  the  fact  that,  of  the  physical 
forces,  some  act  either  directly  at  the  surface  of  a  body,  or  otherwise 
in  proportion  to  its  surface  or  area;  while  others,  and  above  all 
gravity,  act  on  all  particles,  internal  and  external  alike,  and  exert 
a  force  which  is  proportional  to  the  mass,  and  so  usually  to  the 
volume  of  the, body. 

A  simple  case  is  that  of  two  similar  weights  hung  by  two  similar 
wires.  The  forces  exerted  by  the  weights  are  proportional  to  their 
masses,  and  these  to  their  volumes,  and  so  to  the  cubes  of  the 
several  Hnear  dimensions,  including  the  diameters  of  the  wires. 
But  the  areas  of  cross-section  of  the  wires  are  as  the  squares  of  the 
said  linear  dimensions;  therefore  the  stresses  in  the  wires  'per  unit 
area  are  not  identical,  but  increase  in  the  ratio  of  the  linear  dimen- 
sions, and  the  larger  the  structure  the  more  severe  the  strain  becomes : 

Force        l^ 

A^  ^  r^  ^  ^' 

and  the  less  the  wires  are  capable  of  supporting  it. 

In  short,  it  often  happens  that  of  the  forces  in  action  in  a  system 
some  vary  as  one  power  and  some  as  another,  of  the  masses,  distances 
or  other  magnitudes  involved;  the  "dimensions"  remain  the  same 
in  our  equations  of  equilibrium,  but  the  relative  values  alter  with 
the  scale.  This  is  known  as  the  "Principle  of  Similitude,"  or  of 
dynamical  similarity,  and  it  and  its  consequences  are  of  great 
importance.  In  a  handful  of  matter  cohesion,  capillarity,  chemical 
affinity,  electric  charge  are  all  potent;  across  the  solar  system 
gravitation*  rules  supreme;  in  the  mysterious  region  of  the  nebulae, 
it  may  haply  be  that  gravitation  grows  negligible  again. 

To  come  back  to  homelier  things,  the  strength  of  an  iron  girder 
obviously  varies  with  the  cross-section  of  its  members,  and  each 
cross-section  varies  as  the  square  of  a  linear  dimension;  but  the 
weight  of  the  whole  structure  varies  as  the  cube  of  its  linear  dimen- 

*  In  the  early  days  of  the  theory  of  gravitation,  it  was  deemed  especially 
remarkable  that  the  action  of  gravity  "is  proportional  to  the  quantity  of  solid 
matter  in  bodies,  and  not  to  their  surfaces  as  is  usual  in  mechanical  causes;  this 
power,  therefore,  seems  to  surpass  mere  mechanism"  (Colin  Maclaurin,  on  Sir 
Isaac  Newton's  Philosophical  Discoveries,  iv,  9). 


26  ON  MAGNITUDE  [ch. 

sions.  It  follows  at  once  that,  if  we  build  two  bridges  geometrically 
similar,  the  larger  is  the  weaker  of  the  two*,  and  is  so  in  the  ratio 
of  their  linear  dimensions.  It  was  elementary  engineering  experience 
such  as  this  that  led  Herbert  Spencer  to  apply  the  principle  of 
simihtude  to  biologyf. 

But  here,  before  w^e  go  further,  let  us  take  careful  note  that 
increased  weakness  is  no  necessary  concomitant  of  increasing  size. 
There  are  exceptions  to  the  rule,  in  those  exceptional  cases  where  we 
have  to  deal  only  with  forces  which  vary  merely  with  the  area  on 
which  they  impinge.  -  If  in  a  big  and  a  httle  ship  two  similar  masts 
carry  two  similar  sails,  the  two  sails  will  be  similarly  strained,  and 
equally  stressed  at  homologous  places,  and  alike  suitable  for  resisting 
the  force  of  the  same  wind.  Two  similar  umbrellas,  however 
differing  in  size,  will  serve  ahke  in  the  same  weather;  and  the 
expanse  (though  not  the  leverage)  of  a  bird's  wing  may  be  enlarged 
with  little  alteration. 

The  principle  of  similitude  had  been  admirably  apphed  in  a  few 
clear  instances  by  Lesage J,  a  celebrated  eighteenth-century  physician, 
in  an  unfinished  and  unpublished  work.  Lesage  argued,  for  example, 
that  the  larger  ratio  of  surface  to  mass  in  a  small  animal  would  lead 
to  excessive  transpiration,  were  the  skin  as  "porous"  as  our  own; 
and  that  we  may  thus  account  for  the  hardened  or  thickened  skins 
of  insects  and  many  other  small  terrestrial  animals.  Again,  since 
the  weight  of  a  fruit  increases  as  the  cube  of  its  linear  dimensions, 
while  the  strength  of  the  stalk  increases  as  the  square,  it  follows 
that  the  stalk  must  needs  grow  out  of  apparent  due  proportion  to 
the  fruit:    or,  alternatively,  that  tall  trees  should  not  bear  large 

*  The  subject  is  treated  from  the  engineer's  point  of  view  by  Prof.  James 
Thomson,  Comparison  of  similar  structures  as  to  elasticity,  strength  and  stability, 
Coll.  Papers,  1912,  pp.  361-372,  and  Trans.  Inst.  Engineers,  Scotland,  1876;  also 
by  Prof.  A.  Barr,  ibid.  1899.  See  also  Rayleigh,  Nature,  April  22,  1915;  Sir  G. 
Greenhill,  On  mechanical  similitude,  Math.  Gaz.  March  1916,  Coll.  Works,  vi, 
p.  300.  For  a  mathematical  account,  sec  (e.g.)  P.  VV.  Bridgeman,  Dimensional 
Analysis  (2nd  ed.),  1931,  or  F.  W.  Lanchester,  The  Theory  of  Dimensions,  1936. 

t  Herbert  Spencer,  The  form  of  the  earth,  etc.,  Phil.  Mag.  xxx,  pp.  194-6, 
1847;   also  Principles  of  Biology,  pt.  ii,  p.  123  seq.,  1864. 

I  See  Pierre  Prevost,  Notices  de  la  vie  et  des  ecrits  de  Lesage,  1805.  George 
Louis  Lesage,  born  at  Geneva  in  1724,  devoted  sixty-three  years  of  a  life  of  eighty 
to  a  mechanical  theory  of  gravitation;  see  W.  Thomson  (Lord  Kelvin),  On  the 
ultramundane  corpuscles  of  Lesage,  Proc.  E.S.E.  vii,  pp.  577-589,  1872;  Phil.  Mag. 
XLV,  pp.  321-345,  1873;  and  Clerk  Maxwell,  art.  "Atom,"  Encyd.  Brit.  (9),  p.' 46. 


II]  THE  PRINCIPLE  OF  SIMILITUDE  27 

fruit  on  slender  branches,  and  that  melons  and  pumpkins  must  lie 
upon  the  ground.  And  yet  again,  that  in  quadrupeds  a  large  head 
must  be  supported  on  a  neck  which  is  either  excessively  thick  and 
strong  like  a  bull's,  or  very  short  like  an  elephant's*. 

But  it  was  Gahleo  who,  wellnigh  three  hundred  years  ago,  had 
first  laid  down  this  general  principle  of  simiUtude;  and  he  did  so 
with  the  utmost  possible  clearness,  and  with  a  great  wealth  of  illustra- 
tion drawn  from  structures  living  and  deadf.  He  said  that  if  we 
tried  building  ships,  palaces  or  temples  of  enormous  size,  yards, 
beams  and  bolts  would  cease  to  hold  together;  nor  can  Nature 
grow  a  tree  nor  construct  an  animal  beyond  a  certain  size,  while 
retaining  the  proportions  and  employing  the  materials  which  suffice 
in  the  case  of  a  smaller  structure  J.  The  thing  will  fall  to  pieces  of 
its  own  weight  unless  we  either  change  its  relative  proportions,  which 
will  at  length  cause  it  to  become  clumsy,  monstrous  and  inefficient, 
or  else  we  must  find  new  material,  harder  and  stronger  than  was 
used  before.  Both  processes  are  famihar  to  us  in  Nature  and  in 
art,  and  practical  apphcations,  undreamed  of  by  Gahleo,  meet  us  at 
every  turn  in  this  modern  age  of  cement  and  steel  §. 

Again,  as  Galileo  was  also  careful  to  explain,  besides  the  questions 
of  pure  stress  and  strain,  of  the  strength  of  muscles  to  hft  an 
increasing  weight  or  of  bones  to  resist  its  crushing  stress,  we  have 
the  important  question  of  bending  ynornents.  This  enters,  more  or 
less,  into  our  whole  range  of  problems ;  it  aifects  the  whole  form  of 
the  skeleton,  and  sets  a  limit  to  the  height  of  a  tall  tree||. 

*  Cf.  W.  Walton,  On  the  debility  of  large  animals  and  trees,  Quart.  Journ. 
of  Math.  IX,  pp.  179-184,  1868;  also  L.  J.  Henderson,  On  volume  in  Biology, 
Proc.  Amer.  Acad.  Sci.  ii,  pp.  654-658,  1916;   etc. 

t  Discorsi  e  Dimostrazioni  matematiche,  intorno  a  due  nuove  scienze  attenenti 
alia  Mecanica  ed  ai  Muovimenti  Locali:  appresso  gli  Elzevirii,  1638;  Opere, 
ed.  Favaro,  viir,  p.  169  seq.     Transl.  by  Henry  Crew  and  A.  de  Salvio,  1914,  p.  130. 

X  So  Werner  remarked  that  Michael  Angelo  and  Bramanti  could  not  have  built 
of  gypsum  at  Paris  on  the  scale  they  built  of  travertin  at  Rome. 

§  The  Chrysler  and  Empire  State  Buildings,  the  latter  1048  ft.  high  to  the  foot 
of  its  200  ft.  "mooring  mast,"  are  the  last  word,  at  present,  in  this  brobdingnagian 
architecture. 

II  It  was  Euler  and  Lagrange  who  first  shewed  (about  1776-1778)  that  a  column 
of  a  certain  height  would  merely  be  compressed,  but  one  of  a  greater  height  would 
be  bent  by  its  own  weight.  See  Euler,  De  altitudine  columnarum  etc..  Acta  Acad. 
Sci.  Imp.  Petropol.  1778,  pp.  163-193;  G,  Greenhill,  Determination  of  the  greatest 
height  to  which  a  tree  of  given  proportions  can  grow,  Cambr.  Phil.  Soc.  Proc.  rv, 
p.  65,  1881,  and  Chree,  ibid,  vu,  1892. 


28  ON  MAGNITUDE  [ch. 

We  learn  in  elementary  mechanics  the  simple  case  of  two  similar 
beams,  supported  at  both  ends  and  carrying  no  other  weight  than 
their  own.  Within  the  limits  of  their  elasticity  they  tend  to  be 
deflected,  or  to  sag  downwards,  in  proportion  to  the  squares  of  their 
linear  dimensions ;  if  a  match-stick  be  two  inches  long  and  a  similar 
beam  six  feet  (or  36  times  as  long),  the  latter  will  sag  under  its  own 
weight  thirteen  hundred  times  as  much  as  the  other.  To  counteract 
this  tendency,  as  the  size  of  an  animal  increases,  the  limbs  tend  to 
become  thicker  and  shorter  and  the  whole  skeleton  bulkier  and 
heavier;  bones  make  up  some  8  per  cent,  of  the  body  of  mouse  or  wren, 
13  or  14  per  cent,  of  goose  or  dog,  and  17  or  18  per  cent,  of  the  body 
of  a  man.  Elephant  and  hippopotamus  have  grown  clumsy  as  well  as 
big,  and  the  elk  is  of  necessity  less  graceful  than  the  gazelle.  It  is  of 
high  interest,  on  the  other  hand,  to  observe  how  little  the  skeletal 
proportions  differ  in  a  httle  porpoise  and  a  great  whale,  even  in  the 
limbs  and  hmb-bones ;  for  the  whole  influence  of  gravity  has  become 
neghgible,  or  nearly  so,  in  both  of  these. 

In  ifhe  problem  of  the  tall  tree  we  have  to  determine  the  point 
at  which  the  tree  will  begin  to  bend  under  its  own  weight  if  it  be 
ever  so  little  displaced  from  the  perpendicular*.  In  such  an 
investigation  we  have  to  make  certain  assumptions — for  instance 
that  the  trunk  tapers  uniformly,  and  that  the  sectional  area  of  the 
branches  varies  according  to  some  definite  law,  or  (as  Ruskin 
assumed)  tends  to  be  constant  in  any  horizontal  plane;  and  the 
mathematical  treatment  is  apt  to  be  somewhat  difficult.  But 
Greenhill  shewed,  on  such  assumptions  as  the  above,  that  a  certain 
British  Columbian  pine-tree,  of  which  the  Kew  flag-staff,  which  is 
221  ft.  high  and  21  inches  in  diameter  at  the  base,  was  made,  could 
not  possibly,  by  theory,  have  grown  to  more  than  about  300  ft.  It 
is  very  curious  that  Galileo  had  suggested  precisely  the  same  height 
(ducento  braccie  alta)  as  the  utmost  limit  of  the  altitude  of  a  tree. 
In  general,  as  Greenhill  shewed,  the  diameter  of  a  tall  homogeneous 
body  must  increase  as  the  power  3/2  of  its  height,  which  accounts 
for  the  slender  proportions  of  young  trees  compared  with  the  squat 

*  In  like  manner  the  wheat-straw  bends  over  under  the  weight  of  the  loaded 
ear,  and  the  cat's  tail  bends  over  when  held  erect — not  because,  they  "possess 
flexibility,"  but  because  they  outstrip  the  dimensions  within  which  stable  equi- 
librium is  possible  in  a  vertical  position.  The  kitten's  tail,  on  the  other  hand, 
stands  up  spiky  and  straight. 


II]  OF  THE  HEIGHT  OF  A  TREE  29 

or  stunted  appearance  of  old  and  large  ones*.  In  short,  as  Goethe 
says  in  Dicktung  und  Wahrheit,  "Es  ist  dafiir  gesorgt  dass  die  Baume 
nicht  in  den  Himmel  wachsen." 

But  the  tapering  pine-tree  is  but  a  special  case  of  a  wider  problem. 
The  oak  does  not  grow  so  tall  as  the  pine-tree,  but  it  carries  a  heavier 
load,  and  its  boll,  broad-based  upon  its  spreading  roots,  shews  a 
different  contour.  Smeaton  took  it  for  the  pattern  of  his  Hghthouse, 
and  Eiffel  built  his  great  tree  of  steel,  a  thousand  feet  high,  to  a 
similar  but  a  stricter  plan.  Here  the  profile  of  tower  or  tree  follows, 
or  tends  to  follow,  a  logarithmic  curve,  giving  equal  strength 
throughout,  according  to  a  principle  which  we  shall  have  occasion 
to  discuss  later  on,  when  we  come  to  treat  of  form  and  mechanical 
efficiency  in  the  skeletons  of  animals.  In  the  tree,  moreover, 
anchoring  roots  form  powerful  wind-struts,  and  are  most  de- 
veloped opposite  to  the  direction  of  the  prevailing  winds;  for  the 
lifetime  of  a  tree  is  affected  by  the  frequency  of  storms,  and  its 
strength  is  related  to  the  wind-pressure  which  it  must  needs  with- 
standf. 

Among  animals  we  see,  without  the  help  of  mathematics  or  of 
physics,  how  small  birds  and  beasts  are  quick  and  agile,  how  slower 
and  sedater  movements  come  with  larger  size,  and  how  exaggerated 
bulk  brings  with  it  a  certain  clumsiness,  a  certain  inefficiency,  an 
element  of  risk  and  hazard,  a  preponderance  of  disadvantage.  The 
case  was  well  put  by  Owen,  in  a  passage  which  has  an  interest  of 
its  own  as  a  premonition,  somewhat  Hke  De.  Candolle's,  of  the 
"struggle  for  existence."  Owen  wrote  as  follows  J:  "  In  proportion 
to  the  bulk  of  a  species  is  the  difficulty  of  the  contest  which,  as  a 
living  organised  whole,  the  individual  of  each  species  has  to  maintain 
against  the  surrounding  agencies  that  are  ever  tending  to  dissolve 
the  vital  bond,  and  subjugate  the  Hving  matter  to  the  ordinary 
chemical  and  physical  forces.  Any  changes,  therefore,  in  such 
external  conditions  as  a  species  may  have  been  original|y  adapted 

*  The  stem  of  the  giant  bamboo  may  attain  a  height  of  60  metres  while  not  more 
than  about  40  cm.  in  diameter  near  its  base,  which  dimensions  fall  not  far  short 
of  the  theoretical  limits;   A.  J.  Ewart,  Phil.  Trans,  cxcviii,  p.  71,  1906. 

t  Cf.  {int.  al.)  T.  Fetch,  On  buttress  tree-roots,  Ann.  R.  Bot.  Garden,  Peradenyia, 
XI,  pp.  277-285,  1930.  Also  au  interesting  paper  by  James  Macdonald,  on  The 
form  of  coniferous  trees.  Forestry,  vi,  1  and  2,  1931/2. 

X  Trans.  Zool.  Soc.  iv,  p.  27,  1850. 


30  ON  MAGNITUDE  [ch. 

to  exist  in,  will  militate  against  that  existence  in  a  degree 
proportionate,  perhaps  in  a  geometrical  ratio,  to  the  bulk  of  the 
species.  If  a  dry  season  be  greatly  prolonged,  the  large  mammal 
will  suffer  from  the  drought  sooner  than  the  small  one;  if  any 
alteration  of  climate  affect  the  quantity  of  vegetable  food,  the 
bulky  Herbivore  will  be  the  first  to  feel  the  effects  of  stinted 
nourishment." 

But  the  principle  of  GaHleo  carries  us  further  and  along  more 
certain  lines.  The  strength  of  a  muscle,  like  that  of  a  rope  or 
girder,  varies  with  its  cross-section;  and  the  resistance  of  a  bone 
to  a  crushing  stress  varies,  again  hke  our  girder,  with  its  cross- 
section.  But  in  a  terrestrial  animal  the  weight  which  tends  to 
crush  its  limbs,  or  which  its  muscles  have  to  move,  varies  as  the 
cube  of  its  hnear  dimensions;  and  so,  to  the  possible  magnitude 
of  an  animal,  living  under  the  direct  action  of  gravity,  there  is  a 
definite  limit  set.  The  elephant,  in  the  dimensions  of  its  limb-bones, 
is  already  shewing  signs  of  a  tendency  to  disproportionate  thickness 
as  compared  with  the  smaller  mammals;  its  movements  are  in 
many  ways  hampered  and  its  agility  diminished:  it  is  already 
tending  towards  the  maximal  Hmit  of  size  which  the  physical  forces 
permit*.  The  spindleshanks  of  gnat  or  daddy-long-legs  have  their 
own  factor  of  safety,  conditional  on  the  creature's  exiguous  bulk 
and  weight;  for  after  their  own  fashion  even  these  small  creatures 
tend  towards  an  inevitable  limitation  of  their  natural  size.  But,  as 
Gahleo  also  saw,  if  the  animal  be  wholly  immersed  in  water  like  the 
whale,  or  if  it  be  partly  so,  as  was  probably  the  case  with  the  giant 
reptiles  of  the  mesozoic  age,  then  the  weight  is  counterpoised  to 
the  extent  of  an  equivalent  volume  of  water,  and  is  completely 
counterpoised  if  the  density  of  the  animal's  body,  with  the  included 
air,  be  identical  (as  a  whale's  very  nearly  is)  with  that  of  the  water 
around^.  Under  these  circumstances  there  is  no  longer  the  same 
physical  barrier  to  the  indefinite  growth  of  the  animal.  Indeed,  in  the 
case  of  the  aquatic  animal,  there  is,  as  Herbert  Spencer  pointed  out, 


*  Cf.  A.  Rauber,  Galileo  iiber  Knochenformen,  Morphol.  Jahrb.  vii,  p.  327,  1882. 

t  Cf.  W.  S.  Wall,  A  New  Sperm  Whale  etc.,  Sydney,  1851,  p.  64:  "As  for 
the  immense  size  of  Cetacea,  it  evidently  proceeds  from  their  buoyancy  in  the 
medium  in  which  they  live,  and  their  being  enabled  thus  to  counteract  the  force  of 

gravity." 


II]  OF  THE  SPEED  OF  A  SHIP  31 

a  distinct  advantage,  in  that  the  larger  it  grows  the  greater  is  its 
speed.  For  its  available  energy  depends  on  the  mass  of  its  muscles, 
while  its  motion  through  the  water  is  opposed,  not  by  gravity,  but 
by  "skin-friction,"  which  increases  only  as  the  square  of  the  linear 
dimensions*:  whence,  other  things  being  equal,  the  bigger  the  ship 
or  the  bigger  the  fish  the  faster  it  tends  to  go,  but  only  in  the  ratio 
of  the  square  root  of  the  increasing  length.  For  the  velocity  (F) 
which  the  fish  attains  depends  on  the  work  (W)  it  can  do  and  the 
resistance  (R)  it  must  overcome.  Now  we^  have  seen  that  the 
dimensions  of  W  are  l^,  and  of  R  are  l'^ ;  and  by  elementary  mechanics 

WocRV^  or   F^oc^. 

73 

Therefore  V^oCr-=l,  and   F  oc  Vl. 

This  is  what  is  known  as  Fronde's  Law,  of  the  correspondence 
of  speeds — a  simple  and  most  elegant  instance  of  "dimensional 
theory!." 

But  there  is  often  another  side  to  these  questions,  which  makes 
them  too  complicated  to  answer  in  a  word.  For  instance,  the  work 
(per  stroke)  of  which  two  similar  engines  are  capable  should  vary  as 
the  cubes  of  their  linear  dimensions,  for  it  varies  on  the  one  hand 
with  the  area  of  the  piston,  and  on  the  other  with  the  length  of  the 
stroke;  so  is  it  likewise  in  the  animal,  where  the  corresponding 
ratio  depends  on  the  cross-section  of  the  muscle,  and  on  the  distance 
through  which  it  contracts.  But  in  two  similar  engines,  the  available 
horse-power  varies  as  the  square  of  the  linear  dimensions,  and  not 
as  the  cube ;  and  this  for  the  reason  that  the  actual  energy  developed 
depends  on  the  heating-surface  of  the  boiler  J.     So  likewise  must 

*  We  are  neglecting  "drag"  or  "head-resistance,"  which,  increasing  as  the  cube 
of  the  speed,  is  a  formidable  obstacle  to  an  unstreamlined  body.  But  the  perfect 
streamlining  of  whale  or  fish  or  bird  lets  the  surrounding  air  or  water  behave  like 
a  perfect  fluid,  gives  rise  to  no  "surface  of  discontinuity,"  and  the  creature  passes 
through  it  without  recoil  or  turbulence.  Froude  reckoned  skin-friction,  or  surface- 
resistance,  as  equal  to  that  of  a  plane  as  long  as  the  vessel's  water-line,  and  of  area 
equal  to  that  of  the  wetted  surface  of  the  vessel. 

t  Though,  as  Lanchester  says,  the  great  designer  "was  not  hampered  by  a 
knowledge  of  the  theory  of  dimensions." 

X  The  analogy  is  not  a  very  strict  or  complete  one.  We  are  not  taking  account, 
for  instance,  of  the  thickness  of  the  boiler-plates. 


32  ON  MAGNITUDE  [ch. 

there  be  a  similar  tendency  among  animals  for  the  rate  of  supply 
o^  kinetic  energy  to  vary  with  the  surface  of  the  lung,  that  is  to  say 
(other  things  being  equal)  with  the  square  of  the  linear  dimensions 
of  the  animal ;  which  means  that,  caeteris  paribus,  the  small  animal 
is  stronger  (having  more  power  per  unit  weight)  than  a  large  one. 
We  may  of  course  (departing  from  the  condition  of  similarity)  increase 
the  heating-surface  of  the  boiler,  by  means  of  an  internal  system  of 
tubes,  without  increasing  its  outward  dimensions,  and  in  this  very 
way  Nature  increases  the  respiratory  surface  of  a  lung  by  a  complex 
system  of  branching  tubes  and  minute  air-cells ;  but  nevertheless  in 
two  similar  and  closely  related  animals,  as  also  in  two  steam-engines 
of  the  same  make,  the  law  is  bound  to  hold  that  the  rate  of  working 
tends  to  vary  with  the  square  of  the  linear  dimensions,  according  to 
Froude's  law  of  steamship  comparison.  In  the  case  of  a  very  large 
ship,  built  for  speed,  the  difficulty  is  got  over  by  increasing  the  size 
and  number  of  the  boilers,  till  the  ratio  between  boiler-room  and 
engine-room  is  far  beyond  what  is  required  in  an  ordinary  small 
vessel*;  but  though  we  find  lung-space  increased  among  animals 
where  greater  rate  of  working  is  required,  as  in  general  among  birds, 
I  do  not  know  that  it  can  be  shewn  to  increase,  as  in  the  "over- 
boilered"  ship,  with  the  size  of  the  animal,  and  in  a  ratio  which 
outstrips  that  of  the  other  bodily  dimensions.  If  it  be  the  case  then, 
that  the  working  mechanism  of  the  muscles  should  be  able  to  exert 
a  force  proportionate  to  the  cube  of  the  linear  bodily  dimensions, 

*  Let  L  be  the  length,  S  the  (wetted)  surface,  T  the  tonnage,  D  the  displacement 
(or  volume)  of  a  ship;  and  let  it  cross  the  Atlantic  at  a  speed  V.  Then,  in  com- 
paring two  ships,  similarly  constructed  but  of  different  magnitudes,  we  know  that 
L=V\  S=L^  =  V\  D  =  T  =  L^=V^;  also  B  (resistance)  =/Sf.  F^^  F«;  H  (horse- 
power) =  i2 .  F  =  F' ;  and  the  coal  (C)  necessary  for  the  voy a,ge— HjV  =  V^.  That 
is  to  say,  in  ordinary  engineering  language,  to  increase  the  speed  across  the  Atlantic 
by  1  per  cent,  the  ship's  length  must  be  increased  2  per  cent.,  her  tonnage  or 
displacement  6  per  cent.,  her  coal- consumption  also  6  per  cent.,  her  horse-power, 
and  therefore  her  boiler-capacity,  7  per  cent.  Her  bunkers,  accordingly,  keep 
pace  with  the  enlargement  of  the  ship,  but  her  boilers  tend  to  increase  out  of 
proportion  to  the  space  available.  Suppose  a  steamer  400  ft.  long,  of  2000  tons, 
2000  H.P.,  and  a  speed  of  14  knots.  The  corresponding  vessel  of  800  ft.  long  should 
develop  a  speed  of  20  knots  (I  :  2  ::  14^  :  20^),  her  tonnage  would  be  16,000,  her 
H.p.  25,000  or  thereby.  Such  a  vessel  would  probably  be  driven  by  four  propellers 
instead  of  one,  each  carrying  8000  h.p.  See  (int.  al.)  W.  J.  Millar,  On  the  most 
economical  speed  to  drive  a  steamer,  Proc.  Edin.  Math.  Soc.  vii,  pp.  27-29,  1889; 
Sir  James  R.  Napier,  On  the  most  profitable  speed  for  a  fully  laden  cargo  steamer 
for  a  given  voyage,  Proc.  Phil.  Soc,  Glasgow,  vi,  pp.  33-38,  1865. 


II]  OF  FROUDE'S  LAW  33 

while  the  respiratory  mechanism  can  only  supply  a  store  of  energy 
at  a  rate  proportional  to  the  square  of  the  said  dimensions,  the 
singular  result  ought  to  follow  that,  in  swimming  for  instance,  the 
larger  fish  ought  to  be  able  to  put  on  a  spurt  of  speed  far  in  excess 
of  the  smaller  one;  but  the  distance  travelled  by  the  year's  end 
should  be  very  much  ahke  for  both  of  them.  And  it  should  also 
follow  that  the  curve  of  fatigue  is  a  steeper  one,  and  the  staying 
power  less,  in  the  smaller  than  ixx  the  larger  individual.  This  is  the 
c^ase  in  long-distance  racing,  where  neither  draws  far  ahead  until 
the  big  winner  puts  on  his  big  spurt  at  the  end;  on  which  is  based 
an  aphorism  of  the  turf,  that  "a  good  big  'un  is  better  than  a  good 
httle  'un."  For  an  analogous  reason  wise  men  know  that  in  the 
'Varsity  boat-race  it  is  prudent  and  judicious  to  bet  on  the  heavier 
crew. 

Consider  again  the  dynamical  problem  of  the  movements  of  the 
body  and  the  hmbs.  The  work  done  (W)  in  moving  a  Hmb,  whose 
weight  is  p,  over  a  distance  s,  is  measured  hy  ps;  p  varies  as  the 
cube  of  the  hnear  dimensions,  and  s,  in  ordinary  locomotion,  varies 
as  the  linear  dimensions,  that  is  to  say  as  the  length  of  Hmb : 

Wocpsocl^  xl^lK 

But  the  work  done  is  limited  by  the  power  available,  and  this 
varies  as  the  mass  of  the  muscles,  or  as  l^;  and  under  this  hmitation 
neither  p  nor  s  increase  as  they  would  otherwise  tend  to  do.  The 
limbs  grow  shorter,  relatively,  as  the  animal  grows  bigger;  and 
spiders,  daddy-long-legs  and  such-hke  long-limbed  creatures  attain 
no  great  size. 

Let  us  consider  more  closely  the  actual  energies  of  the  body. 
A  hundred  years  ago,  in  Strasburg,  a  physiologist  and  a  mathema- 
tician were  studying  the  temperature  of  warm-blooded  animals*. 
The  heat  lost  must,  they  said,  be  proportional  to  the  surface  of  the 
animal :  and  the  gain  must  be  equal  to  the  loss,  since  the  temperature 
of  the  body  keeps  constant.  It  would  seem,  therefore,  that  the 
heat  lost  by  radiation  and  that  gained  by  oxidation  vary  both  alike, 
as  the  surface-area,  or  the  square  of  the  Hnear  dimensions,  of  the 
animal.     But  this  result  is  paradoxical;    for  whereas  the  heat  lost 

*  MM.  Rameaux  et  Sarrus,  Bull.  Acad.  R.  de  Me'decine,  in,  pp.   1094-1100, 
1838-39. 


34  ON  MAGNITUDE  [ch. 

may  well  vary  as  the  surface-area,  that  produced  by  oxidation 
ought  rather  to  vary  as  the  bulk  of  the  animal:  one  should  vary 
as  the  square  and  the  other  as  the  cube  of  the  Hnear  dimensions. 
Therefore  the  ratio  of  loss  to  gain,  hke  that  of  surface  to  volume, 
ought  to  increase  as  the  size  of  the  creature  diminishes.  Another 
physiologist,  Carl  Bergmann*,  took  the  case  a  step  further.  It  was  he, 
by  the  way,  whp  first  said  that  the  real  distinction  was  not  between 
warm-blooded  and  cold-blooded  animals,  but  between  those  of 
constant  and  those  of  variable  temperature:  and  who  coined  the 
terms  homoeothermic  and  poecilothermic  which  we  use  today.  He 
was  driven  to  the  conclusion  that  the  smaller  animal  does  produce 
more  heat  (per  unit  of  mass)  than  the  large  one,  in  order  to  keep 
pace  with  surface-loss;  and  that  this  extra  heat-production  means 
more  energy  spent,  more  food  consumed,  more  work  donef.  Sim- 
plified as  it  thus  was,  the  problem  still  perplexed  the  physiologists 
for  years  after.  The  tissues  of  one  mammal  are  much  like  those  of 
another.  We  can  hardly  imagine  the  muscles  of  a  small  mammal 
to  produce  more  heat  {caeteris  paribus)  than  those  of  a  large ;  and 
we  begin  to  wonder  whether  it  be  not  nervous  excitation,  rather  than 
quahty  of  muscular  tissue,  which  determines  the  rate  of  oxidation 
and  the  output  of  heat.  It  is  evident  in  certain  cases,  and  may  be 
a  general  rule,  that  the  smaller  animals  have  the  bigger  brains; 
"plus  I'animal  est  petit,"  says  M.  Charles  Richet,  "plus  il  a  des 
echanges  chimiques  actifs,  et  plus  son  cerveau  est  volumineuxj." 
That  the  smaller  animal  needs  more  food  is.  certain  and  obvious. 
The  amount  of  food  and  oxygen  consumed  by  a  small  flying  insect 
is  enormous;    and  bees  and  flies  and  hawkmoths  and  humming- 

*  Carl  Bergmann,  Verhaltnisse  der  Warmeokonomie  der  Tiere  zu  ihrer  Grosse, 
Gottinger  Studien,  i,  pp.  594-708,  1847 — a  very  original  paper. 

t  The  metabolic  activity  of  sundry  mammals,  per  24  hours,  has  been  estimated 
as  follows : 

Weight  (kilo.)       Calories  per  kilo. 
Guinea-pig  0-7  223 

Rabbit  2  58 

Man  70  33 

Horse  600  22 

Elephant  4000  13 

Whale  150000  circa  1-7 

J  Ch.  Richet,  Recherches  de  calorimetrie.  Arch,  de  Physiologie  (3),  vi,  pp.  237-291, 
450-497,  1885.  Cf.  also  an  interesting  historical  account  by  M.  Elie  le  Breton, 
Sur  la  notion  de  "masse  protoplasmique  active":  i.  Probl^mes  poses  par  la 
signification  de  la  loi  des  surfaces,  ibid.  1906,  p.  606. 


II]  OF  BERGMANN'S  LAW  35 

birds  live  on  nectar,  the  richest  and  most  concentrated  of  foods*. 
Man  consumes  a  fiftieth  part  of  his  own  weight  of  food  daily,  but 
a  mouse  will  eat  half  its  own  weight  in  a  day;  its  rate  of  Hving  is 
faster,  it  breeds  faster,  and  old  age  comes  to  it  much  sooner  than 
to  man.  A  warm-blooded  animal  much  smaller  than  a  mouse 
becomes  an  impossibihty;  it  could  neither  obtain  nor  yet  digest  the 
food  required  to  maintain  its  constant  temperature,  and  hence  no 
mammals  and  no  birds  are  as  small  as  the  smallest  frogs  or  fishes. 
The  disadvantage  of  small  size  is  all  the  greater  when  loss  of  heat 
is  accelerated  by  conduction  as  in  the  Arctic,  or  by  convection  as 
in  the  sea.  The  far  north  is  a  home  of  large  birds  but  not  of  small ; 
bears  but  not  mice  five  through  an  Arctic  winter;  the  'least  of  the 
dolphins  Hve  in  warm  waters,  and  there  are  no  small  mammals  in 
the  sea.     This  principle  is  sometimes  spoken  of  as  Bergmann's  Law. 

The  whole  subject  of  the  conservation  of  heat  and  the  maintenance 
of  an  all  but  constant  temperature  in  warm-blooded  animals  interests 
the  physicist  and  the  physiologist  ahke.  It  drew  Kelvin's  attention 
many  years  agof,  and  led  him  to  shew,  in  a  curious  paper,  how 
larger  bodies  are  kept  warm  by  clothing  while  smaller  are  only 
cooled  the  more.  If  a  current  be  passed  through  a  thin  wire,  of 
which  part  is  covered  and  part  is  bare,  the  thin  bare  part  may  glow 
with  heat,  while  convection-currents  streaming  round  the  covered 
part  cool  it  off  and  leave  it  in  darkness.  The  hairy  coat  of  very 
small  animals  is  apt  to  look  thin  and  meagre,  but  it  may  serve  them 
better  than  a  shaggier  covering. 

Leaving  aside  the  question  of  the  supply  of  energy,  and  keeping 
to  that  of  the  mechanical  efficiency  of  the  machine,  we  may  find 
endless  biological  illustrations  of  the  principle  of  simihtude.  All 
through  the  physiology  of  locomotion  we  meet  with  it  in  various 
ways :  as,  for  instance,  when  we  see  a  cockchafer  carry  a  plate  many 
times  its  own  weight  upon  its  back,  or  a  flea  jump  many  inches  high. 
''A  dog,"  says  Gahleo,  "could  probably  carry  two  or  three  such 
dogs  upon  his  back;  but  I  believe  that  a  horse  could  not  carry 
even  one  of  his  own  size." 

♦  Cf.  R.  A.  Davies  and  G.  Fraenkel,  The  oxygen- consumption  of  flies  during 
flight,  Jl.  Exp.  Biol.  XVII,  pp.  402-407,  1940. 

t  W.  Thomson,  On  the  efficiency  of  clothing  for  maintaining  temperature, 
Nature,  xxix,  p.  567,  1884. 


36  ON  MAGNITUDE  [ch. 

Such  problems  were  admirably  treated  by  Galileo  and  Borelli, 
but  many  writers  remained  ignorant  of  their  work.  Linnaeus 
remarked  that  if  an  elephant  were  as  strong  in  proportion  as  a 
stag-beetle,  it  would  be  able  to  pull  up  rocks  and  level  mountains; 
and  Kirby  and  Spence  have  a  well-known  passage  directed  to  shew 
that  such  powers  as  have  been  conferred  upon  the  insect  have  been 
withheld  from  the  higher  animals,  for  the  reason  that  had  these 
latter  been  endued  therewith  they  would  have  "caused  the  early 
desolation  of  the  world*." 

Such  problems  as  that  presented  by  the  flea's  jumping  powersf , 
though  essentially  physiological  in  their  nature,  have  their  interest 
for  us  here:  because  a  steady,  progressive  diminution  of  activity 
with  increasing  size  would  tend  to  set  Hmits  to  the  possible  growth 
in  magnitude  of  an  animal  just  as  surely  as  those  factors  which 
tend  to  break  and  crush  the  living  fabric  under  its  own  weight.  In 
the  case  of  a  leap,  we  have  to  do  rather  with  a  sudden  impulse  than 
with  a  continued  strain,  and  this  impulse  should  be  measured  in 
terms  of  the  velocity  imparted.  The  velocity  is  proportional  to 
the  impulse  (x),  and  inversely  proportional  to  the  mass  (M)  moved : 
V  =  xjM.  But,  according  to  what  we  still  speak  of  as  "Borelh's 
law,"  the  impulse  (i.e.  the  work  of  the  impulse)  is  proportional  to 
the  volume  of  the  muscle  by  which  it  is  produced  {,  that  is  to  say 
(in  similarly  constructed  animals)  to  th^  mass  of  the  whole  body; 
for  the  impulse  is  proportional  on  the  one  hand  to  the  cross-section 
of  the  muscle,  and  on  the  other  to  the  distance  through  which  it 

*  Introduction  to  Entomology,  ii,  p.  190,  1826.  Kirby  and  Spence,  like  many  less 
learned  authors,  are  fond  of  popular  illustrations  of  the  "wonders  of  Nature," 
to  the  neglect  of  dynamical  principles.  They  suggest  that  if  a  white  ant  were  as 
big  as  a  man,  its  tunnels  would  be  "magnificent  cylinders  of  more  than  three 
hundred  feet  in  diameter";  and  that  if  a  certain  noisy  Brazilian  insect  were  as 
big  as  a  man,  its  voice  would  be  heard  all  the  world  over,  "so  that  Stentor  becomes 
a  mute  when  compared  with  these  insects!"  It  is  an  easy  consequence  of 
anthropomorphism,  and  hence  a  common  characteristic  of  fairy-tales,  to  neglect 
the  dynamical  and  dwell  on  the  geometrical  aspect  of  similarity. 

•f  The  flea  is  a  very  clever  jumper;  he  jumps  backwards,  is  stream-lined  ac- 
cordingly, and  alights  on  his  two  long  hind-legs.  Cf.  G.  I.  Watson,  in  Nature, 
21  May  1938. 

X  That  is  to  say,  the  available  energy  of  muscle,  in  ft.-lbs.  per  lb.  of  muscle,  is 
the  same  for  all  animals:  a  postulate  which  requires  considerable  qualification 
when  we  come  to  compare  very  different  kinds  of  muscle,  such  as  the  insect's  and 
the  mammal's. 


II]  OF  BORELLI'S  LAW  37 

contracts.  It  follows  from  this  that  the  velpcity  is  constant,  what- 
ever be  the  size  of  the  animal. 

Putting  it  still  more  simply,  the  work  done  in  leaping  is  propor- 
tional to  the  mass  and  to  the  height  to  which  it  is  raised,  W  oc  mH. 
But  the  muscular  power  available  for  this  work  is  proportional  to 
the  mass  of  muscle,  or  (in  similarly  constructed  animals)  to  the  mass 
of  the  animal,  W  oc  m.  It  follows  that  H  is,  or  tends  to  be,  a 
constant.  In  other  words,  all  animals,  provided  always  that  they 
are  similarly  fashioned,  with  their  various  levers  in  Hke  proportion, 
ought  to  jump  not  to  the  same  relative  but  to  the  same  actual 
height*.  The  grasshopper  seems  to  be  as  well  planned  for  jumping 
as  the  flea,  and  the  actual  heights  to  which  they  jump  are  much  of 
a  muchness;  but  the  flea's  jump  is  about  200  times  its  own  height, 
the  grasshopper's  at  most  20-30  times;  and  neither  flea  nor  grass- 
hopper is  a  better  but  rather  a  worse  jumper  than  a  horse  or  a  manf . 

As  a  matter  of  fact,  Borelli  is  careful  to  point  out  that  in  the  act 
of  leaping  the  impulse  is  not  actually  instantaneous,  like  the  blow 
of  a  hammer,  but  takes  some  httle  time,  during  which  the  levers 
are  being  extended  by  which  the  animal  is  being  propelled  forwards ; 
and  this  interval  of  time  will  be  longer  in  the  case  of  the  longer 
levers  of  the  larger  animal.  To  some  extent,  then,  this  principle 
acts  as  a  corrective  to  the  more  general  one,  and  tends  to  leave  a 
certain  balance  of  advantage  in  regard  to  leaping  power  on  the  side 
of  the  larger  animal  J.  But  on  the  other  hand,  the  question  of 
strength  of  materials  comes  in  once  more,  and  the  factors  of  stress 
and  strain  and  bending  moment  make  it  more  and  more  difficult 
for  nature  to  endow  the  larger  animal  with  the  length  of  lever  with 
which  she  has  provided  the  grasshopper  or  the  flea.  To  Kirby  and 
Spence  it  seemed  that  "This  wonderful  strength  of  insects  is 
doubtless  the  result  of  something  peculiar  in  the  structure  and 
arrangement  of  their  muscles,  and  principally  their  extraordinary 

*  Borelli,  Prop.  CLxxvn.  Animalia  minora  et  minus  ponderosa  majores  saltus 
efficiunt  respectu  sui  corporis,  si  caetera  fuerint  paria. 

t  The  high  jump  is  nowadays  a  highly  skilled  performance.  For  the  jumper 
contrives  that  his  centre  of  gravity  goes  under  the  bar,  while  his  body,  bit  by  bit, 
goes  over  it. 

J  See  also  {int.  al.),  John  Bernoulli,  De  Motu  Musculorum,  Basil.,  1694; 
Chabry,  Mecanisme  du  saut,  J.  de  VAnat.  et  de  la  Physiol,  xix,  1883;*  Sur  la 
longueur  des  membres  des  animaux  sauteurs,  ibid,  xxi,  p.  356,  1885;  Le  Hello, 
De  Taction  des  organes  locomoteurs,  etc.,  ibid,  xxix,  pp.  65-93,  1893;  etc. 


38  ON  MAGNITUDE  [ch. 

power  of  contraction."  This  hypothesis,  which  is  so  easily  seen  on 
physical  grounds  to  be  unnecessary,  has  been  amply  disproved  in 
a  series  of  excellent  papers  by  Felix  Plateau*. 

From  the  impulse  of  the  preceding  case  we  may  pass  to  the  momentum 
created  (or  destroyed)  under  similar  circumstances  by  a  given  force  acting 
for  a  given  time :  mv  =  Ft. 

We  know  that    .  m  oc  P,    and    t  =  Ifv, 

so  that  lH  =  Fllv,    or    v^  =  F/l\ 

But  whatsoever  force  be  available,  the  animal  may  only  exert  so  much  of 
it  as  is  in  proportion  to  the  strength  of  his  own  limbs,  that  is  to  say  to  the 
cross-section  of  bone,  sinew  and  muscle ;  and  all  of  these  cross-sections  are 
proportional  to  P,  the  square  of  the  linear  dimensions.  The  maximal  force, 
-f'max,  which  the  animal  dare  exert  is  proportional,  then,  to  l^;  therefore 

Fraa^JP  =  coustant. 

And  the  maximal  speed  which  the  animal  can  safely  reach,  namely 
F'max  =  ^max/^,  IS  also  constaut,  or  independent  {ceteris  paribus)  of  the  dimensions 
of  the  animal. 

A  spurt  or  effort  may  be  well  within  the  capacity  of  the  animal  but  far 
beyond  the  margin  of  safety,  as  trainer  and  athlete  well  know.  This  margin 
is  a  narrow  one,  whether  for  athlete  or  racehorse;  both  run  a  constant  risk 
of  overstrain,  under  which  they  may  "pull"  a  muscle,  lacerate  a  tendon,  or 
even  "break  down"  a  bonet. 

It  is  fortunate  for  their  safety  that  animals  do  not  jump  to  heights  pro- 
portional to  their  own.  For  conceive  an  animal  (of  mass  m)  to  jump  to 
a  certain  altitude,  such  that  it  reaches  the  ground  with  a  velocity  v;  then 
if  c  be  the  crushing  strain  at  any  point  of  the  sectional  area  (A)  of  the  limbs, 
the  limiting  condition  is  that  mv  =  cA. 

If  the  animal  vary  in  magnitude  without  change  in  the  height  to  which 
it  jumps  (or  in  the  velocity  with  which  it  descends),  then 

m       P  , 

coc  2  oc  ^"2,     orZ. 

The  crushing  strain  varies  directly  with  the  linear  dimensions  of  the  animal ; 
and  this,  a  dynamical  case,  is  identical  with  the  usual  statical  limitalfion  of 
magnitude. 

*  Recherches  sur  la  force  absolue  des  muscles  des  Inverlebres,  Bull.  Acad.  R, 
de  Belgique  (3),  vi,  vii,  1^83-84:  see  also  ibid.  (2),  xx,  1865;  xxii,  1866;  Ann. 
Mag.  N.H.  xvii,  p.  139,  1866;  xix,  p.  95,  1867.  Cf.  M.  Radau,  Sur  la  force 
musculaire  des  insectes,  Revue  des  deux  Mondes,  lxiv,  p.  770,  1866.  The  subject 
had  been  well  treated  by  Straus-Diirckheim,  in  his  Considerations  generates  sur 
Vanatomie  comparee  des  animauz  articuUs,  1828. 

t  Cf.  The  dynamics  of  sprint -running,  by  A.  V.  Hill  and  others,  Proc.  R.S.  (B), 
cii,  pp.  29-42,  1927;  or  Muscular  Movement  in  Man,  by  A.  V.  Hill,  New  York, 
1927,  ch.  VI,  p.  41. 


II]  OF  LOCOMOTION  39 

But  if  the  animal,  with  increasing  size  or  stature,  jump  to  a  correspondingly 
increasing  height,  the  case  becomes  much  more  serious.  For  the  final  velocity 
of  descent  varies  as  the  square  root  of  the  altitude  reached,  and  therefore  as 
the  square  root  of  tte  linear  dimensions  of  the  animal.     And  since,  as  before, 

13 

cccmvcc  j^  Vy 

c  oc  75 .  V ;,    or  c  oc  Z». 

If  a  creature's  jump  were  in  proportion  to  its  height,  the  crushing  strains 
would  so  increase  that  its  dimensions  would  be  limited  thereby  in  a  much  higher 
degree  than  was  indicated  by  statical  considerations.  An  animal  may  grow 
to  a  size  where  lit  is  unstable  dynamically,  though  still  on  the  safe  side 
statically — a  size  where  it  moves  with  difficulty  though  it  rests  secure.  It  is 
by  reason  of  dynamical  rather  than  of  statical  relations  that  an  elephant 
is  of  graver  deportment  than  a. mouse. 

An  apparently  simple  problem,  much  less  simple  than  it  looks,  lies 
in  the  act  of  walking,  where  there  will  evidently  be  great  economy  of 
work  if  the  leg  swing  with  the  help  of  gravity,  that  is  to  say,  at  a 
peTidulum-rate.  The  conical  shape  and  jointing  of  the  limb,  the  time 
spent  with  the  foot  upon  the  ground,  these  and  other  mechanical 
differences  complicate  the  case,  and  make  the  rate  hard  to  define  or 
calculate.  Nevertheless,  we  may  convince  ourselves  by  counting  our 
steps,  that  the  leg  does  actually  tend  to  swing,  as  a  pendulum  does, 
at  a  certain  definite  rate*.  So  on  the  same  principle,  but  to  the 
slower  beat  of  a  longer  pendulum,  the  scythe  swings  smoothly  in 
the  mower's  hands. 

To  walk  quicker,  we  "step  out";  we  cause  the  leg-pendulum  to 
describe  a  greater  arc,  but  it  does  not  swing  or  vibrate  faster  until 
we  shorten  the  pendulum  and  begin  to  run.  Now  let  two  similar 
individuals,  A  and  B,  walk  in  a  similar  fashion,  that  is  to  say  with 
a  similar  an^le  of  swing  (Fig.  1).  The  arc  through  which  the  leg 
swings,  or  the  amplitude  of  each  step,  will  then  vary  as  the  length 
of  leg  (say  as  a/6),  and  so  as  the  height  or  other  linear  dimension  (/) 
of  the  manf.     But  the  time  of  swing  varies  inversely  as'the  square 

*  The  assertion  that  the  hmb  tends  to  swing  in  pendulum-time  was  first  made 
by  the  brothers  Weber  {Mechanik  der  menschl.  Gehwerkzeuge,  Gottingen,  1836). 
Some  later  writers  have  criticised  the  statement  (e.g.  Fischer,  Die  Kinematik  dea 
Beinschwingens  etc.,  AhJi.  math.  phys.  Kl.  k.  Sachs.  Ges.  xxv-xxvin,  1899-1903), 
but  for  all  that,  with  proper  and  large  qualifications,  it  remains  substantially  true. 

t  So  the  stride  of  a  Brobdingnagian  was  10  yards  long,  or  just  twelve  times  the 
2  ft.  6  in.,  which  make  the  average  stride  or  half-pace  of  a  man. 


40 


ON  MAGNITUDE 


[CH. 


root  of  the  pendulum-length,  or  Va/Vb.     Therefore  the  velocity, 
which  is  measured  by  amphtude/time,  or  a/b  x  Vb/Va,  will  also  vary 

as  the  square  root  of  the  hnear  dimen- 
sions; which  is  Froude's  law  over  again. 
The  smaller  man,  or  smaller  animal, 
goes  slower  than  the  larger,  but  only  in 
the  ratio  of  the  square  roots  of  their 
linear  dimensions ;  whereas,  if  the  limbs 
moved  alike,  irrespective  of  the  size  of 
the  animal — if  the  limbs  of  the  mouse 
swung  no  faster  than  those  of  the  horse 
— then  the  mouse  would  be  as  slow  in 
its  gait  or  slower  than  the  tortoise. 
M.  DeHsle*  saw  a  fly  walk  three  inches 
in  half-a-second ;  this  was  good  steady 
walking.  When  we  walk  five  miles  an 
hour  we  go  about  88  inches  in  a  second, 
or  88/6  =14-7  times  the  pace  of  M.  DeHsle 's  fly.  We  should  walk 
at  just  about  the  fly's  pace  if  our  stature  were  1/(14-7)2,  or  1/216 
of  our  present  height — say  72/216  inches,  or  one-third  of  an  inch 
high.  Let  us  note  in  passing  that  the  number  of  legs  does  not 
matter,  any  more  than  the  number  of  wheels  to  a  coach;  the 
centipede  runs  none  the  faster  for  all  his  hundred  legs. 

But  the  leg  comprises  a  complicated  system  of  levers,  by  whose 
various  exercise  we  obtain  very  different  results.  For  instance,  by 
being  careful  to  rise  upon  our  instep  we  increase  the  length  or 
amplitude  of  our  stride,  and  improve  our  speed  very  materially; 
and  it  is  curious  to  see  how  Nature  lengthens  this  metatarsal  joint, 
or  instep-lever,  in  horse f  and  hare  and  greyhound,  in  ostrich  and 
in  kangaroo,  and  in  every  speedy  animal.  Furthermore,  in  running 
we  bend  and  so  shorten  the  leg,  in  order  to  accommodate  it  to  a 
quicker  rate  of  pendulum-swing  J.    In  short  the  jointed  structure 

*  Quoted  in  Mr  John  Bishop's  interesting  article  in  Todd's  Cyclopaedia,  iii, 
p.  443. 

t  The  "cannon-bones"  are  not  only  relatively  longer  but  may  even  be  actually 
longer  in  a  little  racehorse  than  a  great  carthorse. 

t  There  is  probably  another  factor  involved  here:  for  in  bending  and  thus 
shortening  the  leg,  we  bring  its  centre  of  gravity  nearer  to  the  pivot,  that  is  to 
say  to  the  joint,  and  so  the  muscle  tends  to  move  it  the  ,more  quickly.     After  all, 


II]  OF  RATE  OF  WALKING  41 

of  the  leg  permits  us  to  use  it  as  the  shortest  possible  lever  while 
it  is  swinging,  and  as  the  longest  possible  lever  when  it  is  exerting 
its  propulsive  force. 

The  bird's  case  is  of  peculiar  interest.  In  running,  walking  or 
swimming,  we  consider  the  speed  which  an  animal  can  attain,  and 
the  increase  of  speed  which  increasing  size  permits  of.  But  in  flight 
there  is  a  certain  necessary  speed— a  speed  (relative  to  the  air)  which 
the  bird  must  attain  in  order  to  maintain  itself  aloft,  and  which  nfiust 
increase  as  its  size  increases.  It  is  highly  probable,  as  Lanchester 
remarks,  that  Lilienthal  met  his  untimely  death  (in  August  1896) 
not  so  much  from  any  intrinsic  fault  in  the  design  or  construction 
of  his  machine,  but  simply  because  his  engine  fell  somewhat 
short  of  the  power  required  to  give  the  speed  necessary  for  its 
stability. 

Twenty-five  years  ago,  when  this  book  was  written,  the  bird,  or 
the  aeroplane,  was  thought  of  as  a  machine  whose  sloping  wings, 
held  at  a  given  angle  and  driven  horizontally  forward,  deflect  the 
air  downwards  and  derive  support  from  the  upward  reaction.  In 
other  words,  the  bird  was  supposed  to  communicate  to  a  mass  of 
air  a  downward  momentum  equivalent  (in  unit  time)  to  its  own 
weight,  and  to  do  so  by  direct  and  continuous  impact.  The  down- 
ward momentum  is  then  proportional  to  the  mass  of  air  thrust 
downwards,  and  to  the  rate  at  which  it  is  so  thrust  or  driven:  the 
mass  being  proportional  to  the  wing-area  and  to  the  speed  of  the 
bird,  and  the  rate  being  again  proportional  to  the  flying  speed;  so 
that  the  momentum  varies  as  the  square  of  the  bird's  linear  dimen- 
sions and  also  as  the  square  of  its  speed.  But  in  order  to  balance 
its  weight,  this  momentum  must  also  be  proportional  to  the 
cube  of  the  bird's  linear  dimensions;  therefore  the  bird's  necessary 
speed,  such  as  enables  it  to  maintain  level  flight,  must  be  pro- 
portional to  the  square  root  of  its  linear  dimensions,  and  the  whole 
work  done  must  be  proportional  to  the  power  3 J  of  the  said  linear 
dimensions. 

The  case  stands,  so  far,  as  follows :  m,  the  mass  of  air  deflected 
downwards;  M,  the  momentum  so  communicated;  W,  the  work 
done — all  in  unit  time;   w,  the  weight,  and  F,  the  velocity  of  the 

we  know  that  the  pendulum  theory  is  not  the  whole  story,  but  only  an  important 
first  approximation  to  a  complex  phenomenon. 


42  ON  MAGNITUDE  [ch. 

bird;  I,  a  linear  dimension,  the  form  of  the  bird  being  supposed 
constant.       j^j  _  ^^  ^  ^3^  but  M  =  mV ,  and  m  =  W , 

Therefore  M  -  ^272  _  ^3^ 

and  therefore  F  =  VI 

and  Tf  =  MF  =  R 

The  gist  of  the  matter  is,  or  seems  to  be,  that  the  work  which 
can  he  done  varies  with  the  available  weight  of  muscle,  that  is  to  say, 
with  the  mass  of  the  bird ;  but  the  work  which  has  to  be  done  varies 
with  mass  and  distance;  so  the  larger  the  bird  grows,  the  greater 
the  disadvantage  under  which  all  its  work  is  done*.  The  dispropor- 
tion does  not  seem  very  great  at  first  sight,  but  it  is  quite  enough 
to  tell.  It  is  as  much  as  to  say  that,  every  time  we  double  the 
linear  dimensions  of  the  bird,  the  difficulty  of  flight,  >  or  the  work 
which  must  needs  be  done  in  order  to  fly,  is  increased  in  the  ratio 
of  2^  to  2^*,  or  1  :  V2,  or  say  1:14.  If  we  take  the  ostrich  to  exceed 
the  sparrow  in  linear  dimensions  as  25  :  1,  which  seems  well  within 
the  mark,  the  ratio  would  be  that  between  25^*  and  25^,  or  between 
5'  and  5^;  in  other  words,  flight  would  be  five  times  more  difficult 
for  the  larger  than  for  the  smaller  bird. 

But  this  whole  explanation  is  doubly  inadequate.  For  one  thing, 
it  takes  no  account  of  gliding  flight,  in  which  energy  is  drawn  from 
the  wind,  and  neither  muscular  power  nor  engine  power  are  em- 
ployed; and  we  see  that  the  larger  birds,  vulture,  albatross  or 
solan-goose,  depend  on  gliding  more  and  more.  Secondly,  the  old 
simple  account  of  the  impact  of  the  wing  upon  the  air,  and  the 
manner  in  which  a  downward  momentum  is  communicated  and 
support  obtained,  is  now  known  to  be  both  inadequate  and 
erroneous.  For  the  science  of  flight,  or  aerodynamics,  has  grown 
out  of  the  older  science  of  hydrodynamics;  both  deal  with  the 
special  properties  of  a  fluid,  whether  water  or  air;  and  in  our  case, 
to  be  content  to  think  of  the  air  as  a  body  of  mass  m,  to  which  a 
velocity  v  is  imparted,  is  to  neglect  all  its  fluid  properties.    How  the 

*  This  is  the  result  arrived  at  by  Helmholtz,  Ueber  ein  Theorem  geometrisch- 
ahnliche  Bewegungen  fliissiger  Korper  betrefFend,  nebst  Anwendung  auf  das 
Problem  Luftballons  zu  lenken,  Monatsber.  Akad.  Berlin,  1873,  pp.  501-514.  It  was 
criticised  and  challenged  (somewhat  rashly)  by  K.  Miillenhof,  Die  Grosse  der  Flug- 
flachen  etc.,  PfiUger's  Archiv,  xxxv,  p.  407;   xxxvi,  p.  548,  1885. 


II]  OF  FLIGHT  43 

fish  or  the  dolphin  swims,  and  how  the  bird  flies,  are  up  to  a  certain 
point  analogous  problems ;  and  stream-lining  plays  an  essential  part 
in  both.  But  the  bird  is  much  heavier  than  the  air,  and  the  fish 
has  much  the  same  density  as  the  water,  so  that  the  problem  of 
keeping  afloat  or  aloft  is  negligible  in  the  one,  and  all-important  in 
the  other.  Furthermore,  the  one  fluid  is  highly  compressible,  and 
the  other  (to  all  intents  and  purposes)  incompressible ;  and  it  is  this 
very  difference  which  the  bird,  or  the  aeroplane,  takes  special 
advantage  of,  and  which  helps,  or  even  enables,  it  to  fly. 

It  remains  as  true  as  ever  that  a  bird,  in  order  to  counteract 
gravity,  must  cause  air  to  move  downward  and  obtains  an  upward 
reaction  thereby.  But  the  air  displaced  downward  beneath  the 
wing  accounts  for  a  small  and  varying  part,  perhaps  a  third  perhaps 
a  good  deal  less,  of  the  whole  force  derived;  and  the  rest  is  generated 
above  the  wing,  in  a  less  simple  way.  For,  as  the  air  streams  past 
the  slightly  sloping  wing,  as  smoothly  as  the  stream-lined  form 
and  polished  surface  permit,  it  swirls  round  the  front  or  "leading" 
edge*,  and  then  streams  swiftly  over  the  upper  surface  of  the  wing; 
while  it  passes  comparatively  slowly,  checked  by  the  opposing  slope 
of  the  wing,  across  the  lower  side.  And  this  is  as  much  as  to  say 
that  it  tends  to  be  compressed  below  and  rarefied  above;  in  other 
words,  that  a  partial  vacuum  is  formed  above  the  wing  and  follows 
it  wherever  it  goes,  so  long  as  the  stream-lining  of  the  wing  and  its 
angle  of  incidence  are  suitable,  and  so  long  as  the  bird  travels  fast 
enough  through  the  air. 

The  bird's  weight  is  exerting  a  downward  force  upon  the  air,  in 
one  way  just  as  in  the  other;  and  we  can  imagine  a  barometer 
delicate  enough  to  shew  and  measure  it  as  the  bird  flies  overhead. 
But  to  calculate  that  force  we  should  have  to  consider  a  multitude 
of  component  elements;  we  should  have  to  deal  with  the  stream- 
lined tubes  of  flow  above  and  below,  and  the  eddies  round  the  fore- 
edge  of  the  wing  and  elsewhere;  and  the  calculation  which  was  too 
simple  before  now  becomes  insuperably  difficult.  But  the  principle 
of  necessary  speed  remains  as  true  as  ever.     The  bigger  the  bird 

*  The  arched  form,  or  "dipping  front  edge"  of  the  wing,  and  its  use  in  causing 
a  vacuum  above,  were  first  recognised  by  Mr  H.  F.  Phillips,  who  put  the  idea  into 
a  patent  in  1884.  The  facts  were  discovered  independently,  and  soon  afterwards, 
both  by  Lilienthal  and  Lanchester. 


44  ON  MAGNITUDE  [ch. 

becomes,  the  more  swiftly  must  the  air  stream  over  the  wing 
to  give  rise  to  the  rarefaction  or  negative  pressure  which  is  more 
and  more  required;  and  the  harder  must  it  be  to  fly,  so  long  as 
work  has  to  be  done  by  the  muscles  of  the  bird.  The  general 
principle  is  the  same  as  before,  though  the  quantitative  relation 
does  not  work  out  as  easily  as  it  did.  As  a  matter  of  fact,  there 
is  probably  little  difference  in  the  end;  and  in  aeronautics,  the 
"total  resultant  force"  which  the  bird  employs  for  its  support  is 
said,  e^npirically,  to  vary  as  the  square  of  the  air-speed:  which  is 
then  a  result  analogous  to  Froude's  law,  and  is  just  what  we  arrived 
at  before  in  the  simpler  and  less  accurate  setting  of  the  case. 

But  a  comparison  between  the  larger  and  the  smaller  bird,  like 
all  other  comparisons,  applies  only  so  long  as  the  other  factors  in 
the  case  remain  the  same ;  and  these  vary  so  much  in  the  complicated 
action  of  flight  that  it  is  hard  indeed  to  compare  one  bird  with 
another.  For  not  only  is  the  bird  continually  changing  the  incidence 
of  its  wing,  but  it  alters  the  lie  of  every  single  important  feather; 
and  all  the  ways  and  means  of  flight  vary  so  enormously,  in  big 
wings  and  small,  and  Nature  exhibits  so  many  refinements  and 
"  improvements"  in  the  mechanism  required,  that  a  comparison  based 
on  size  alone  becomes  imaginary,  and  is  little  worth  the  making. 

The  above  considerations  are  of  great  practical  importance  in 
aeronautics,  for  they  shew  how  a  provision  of  increasing  speed  must  ac- 
company every  enlargement  of  our  aeroplanes.  Speaking  generally, 
the  necessary  or  minimal  speed  of  an  aeroplane  varies  as  the  square 
root  of  its  Unear  dimensions;  if  (ceteris  paribus)  we  make  it  four 
times  as  long,  it  must,  in  order  to  remain  aloft,  fly  twice  as  fast  as 
before*.  If  a  given  machine  weighing,  say,  500  lb.  be  stable  at 
40  miles  an  hour,  then  a  geometrically  similar  one  which  weighs, 
say,  a  couple  of  tons  has  its  speed  determined  as  follows : 

W:w::L^:l^::S:l. 

Therefore  L:l::2:l. 

But  V^:v^::L:l 

Therefore  V:v::V2:l  =  1-414  : 1. 

*  G.  H.  Bryan,  Stability  in  Aviation,  1911;  F.  W.  Lanchester,  Aerodynamics, 
1909;  cf.  (int.  al.)  George  Greenhill,  The  Dynamics  of  Mechanical  Flight,  1912; 
F.  W.  Headley,  The  Flight  of  Birds,  and  recent  works. 


II]  OF  NECESSARY  SPEED  45 

That  is  to  say,  the  larger  machine  must  be  capable  of  a  speed  of 
40  X  1-414,  or  about  56|,  miles  per  hour. 

An  arrow  is  a  somewhat  rudimentary  flying-machine;  but  it  is 
capable,  to  a  certain  extent  and  at  a  high  velocity,  of  acquiring 
"stability,"  and  hence  of  actual  flight  after  the  fashion  of  an  aero- 
plane; the  duration  and  consequent  range  of  its  trajectory  are 
vastly  superior  to  those  of  a  bullet  of  the  same  initial  velocity. 
Coming  back  to  our  birds,  and  again  comparing  the  ostrich  with 
the  sparrow,  we  find  we  know  little  or  nothing  about  the  actual 
speed  of  the  latter ;  but  the  minimal  speed  of  the  swift  is  estimated 
at  100  ft.  per  second,  or  even  more — say  70  miles  an  hour.  We 
shall  be  on  the  safe  side,  and  perhaps  not  far  wrong,  to  take  20  miles 
an  hour  as  the  sparrow's  minimal  speed;  and  it  would  then  follow 
that  the  ostrich,  of  25  times  the  sparrow's  linear  dimensions,  would 
have  to  fly  (if  it  flew  at  all)  with  a  minimum  velocity  of  5  x  20, 
or  100  miles  an  hour*. 

The  same  principle  of  necessary  speed,  or  the  inevitable  relation 
between  the  dimensions  of  a  flying  object  and  the  minimum  velocity 
at  which  its  flight  is  stable,  accounts  for  a  considerable  number  of 

*  Birds  have  an  ordinary  and  a  forced  speed.  Meinertzhagen  puts  the  ordinary 
flight  of  the  swift  at  68  m.p.h.,  which  talhes  with  the  old  estimate  of  Athanasius 
Kircher  (Physiologia,  ed.  1680,  p.  65)  of  100  ft.  per  second  for  the  swallow.  Abel 
Chapman  {Retrospect,  1928,  ch.  xiv)  puts  the  gliding  or  swooping  flight  of  the  swift 
at  over  150  m.p.h.,  and  that  of  the  griffon  vulture  at  180  m.p.h.;  but  these  skilled 
fliers  doubtless  far  exceed  the  necessary  minimal  speeds  which  we  are  speaking  of. 
An  airman  flying  at  70  m.p.h.  has  seen  a  golden  eagle  fly  past  him  easily;  but 
even  this  speed  is  exceptional.  Several  observers  agree  in  giving  50  m.p.h.  for 
grouse  and  woodcock,  and  30  m.p.h.  for  starling,  chaffinch,  quail  and  crow.  A 
migrating  flock  of  lapwing  travelled  at  41  m.p.h.,  ten  or  twelve  miles  more  than 
the  usual  speed  of  the  single  bird.  Lanchester,  on  theoretical  considerations, 
estimates  the  speed  of  the  herring  gull  at  26  m.p.h.,  and  of  the  albatross  at  about 
34  miles.  A  tern,  a  very  skilful  flier,  was  seen  to  fly  as  slowly  as  15  m.p.h. 
A  hornet  or  a  large  dragonfly  may  reach  14  or  18  m.p.h.;  but  for  most  insects 
2-4  metres  per  sec,  say  4-9  m.p.h.,  is  a  common  speed  (cf,  A.  Magnan,  Vol. 
des  Insectes,  1834,  p.  72).  The  larger  diptera  are  very  swift,  but  their  speed  is  much 
exaggerated.  A  deerfly  (Cephenomyia)  has  been  said  to  fly  at  400  yards  per  second, 
or  say  800  m.p.h.,  an  impossible  velocity  (Irving  Langmuir,  Science,  March  11,  1938). 
It  would  mean  a  pressure  on  the  fly's  head  of  half  an  atmosphere,  probably  enough 
to  crush  the  fly;  to  maintain  it  would  take  half  a  horsepower;  and  this  would  need 
a  food- consumption  of  1^  times  the  fly's  weight  per  second\  25  m.p.h.  is  a  more 
reasonable  estimate.  The  naturalist  should  not  forget,  though  it  does  not  touch 
our  present  argument,  that  the  aeroplane  is  built  to  the  pattern  of  a  beetle  rather 
than  of  a  bird;  for  the  elytra 'are  not  wings  but  planes.  Cf.  int.  ah,  P.  Amans, 
Geometrie. .  .des  ailes  rigides,  C.R.  Assoc.  Frang.  pour  Vavancem.  des  Sc.  1901. 


46  ON  MAGNITUDE  [ch, 

observed  phenomena.  It  tells  us  why  the  larger  birds  have  a 
marked  difficulty  in  rising  from  the  ground,  that  is  to  say,  in 
acquiring  to  begin  with  the  horizontal  velocity  necessary  for  their 
support;  and  why  accordingly,  as  Mouillard*  and  others  have 
observed,  the  heavier  birds,  even  those  weighing  no  more  than  a 
pound  or  two,  can  be  effectually  caged  in  small  enclosures  open 
to  the  sky.  It  explains  why,  as  Mr  Abel  Chapman  says,  "all 
ponderous  birds,  wild  swans  and  geese,  great  bustard  and  caper- 
cailzie, even  blackcock,  fly  faster  than  they  appear  to  do,"  while 
"light-built  types  with  a  big  wing-areaf,  such  as  herons  and  harriers, 
possess  no  turn  of  speed  at  all."  For  the  fact  is  that  the  heavy 
birds  must  fly  quickly,  or  not  at  all.  It  tells  us  why  very  small 
birds,  especially  those  as  small  as  humming-birds,  and  a  fortiori  the 
still  smaller  insects,  are  capable  of  "stationary  flight,"  a  very  slight 
and  scarcely  perceptible  velocity  relatively  to  the  air  being  sufficient 
for  their  support  and  stabihty.  And  again,  since  it  is  in  all 
these  cases  velocity  relatively  to  the  air  which  we  are  speaking 
of,  we  comprehend  the  reason  why  one  may  always  tell  which 
way  the  wind  blows  by  watching  the  direction  in  which  a  bird  starts 
to  fly. 

The  wing  of  a  bird  or  insect,  like  the  tail  of  a  fish  or  the  blade 
of  an  oar,  gives  rise  at  each  impulsion  to  a  swirl  or  vortex,  which 
tends  (so  to  speak)  to  chng  to  it  and  travel  along  with  it;'  and  the 
resistance  which  wing  or  oar  encounter  comes  much  more  from 
these  vortices  than  from  the  viscosity  of  the  fluid.  J  We  learn  as  a 
corollary  to  this,  that  vortices  form  only  at  the  edge  of  oar  or  wing — 
it  is  only  the  length  and  not  the  breadth  of  these  which  matters. 
A  long  narrow  oar  outpaces  a  broad  one,  and  the  efficiency  of  the 
long,  narrow  wing  of  albatross,  swift  or  hawkmoth  is  so  far  accounted 
for.  From  the  length  of  the  wing  we  can  calculate  approximately 
its  rate  of  swing,  and  more  conjecturally  the  dimensions  of  each 
vortex,  and  finally  the  resistance  or  Ufting  power  of  the  stroke; 
and  the  result  shews  once  again  the  advantages  of  the  small-scale 

*  Mouillard,  L' empire  de  Vair;  essai  d'ornithologie  appliqu4e  a  Vaviationf  1881; 
transl.  in  Annual  Report  of  the  Smithsonian  Institution^  1892. 

t  On  wing-area  in  relation  to  weight  of  bird  see  Lendenfeld  in  Naturw.  Wochenschr. 
Nov.  1904,  transl.  in  Smithsonian  Inst.  Rep.  1904;  also  E.  H.  Hankin,  Animal 
Flight,  1913;  etc. 

X  Cf.  V.  Bjerknes,  Hydrodynamique  physique,  n,  p.  293,  1934. 


II]  OF  MODES  OF  FLIGHT  47 

mechanism,  and  the  disadvantage  under  which  the  larger  machine 
or  larger  creature  hes. 


Length 

Speed  of 

Radius 

Force  of 

Specific 

Weight 

of  wing     Beats 

wing-tip 

of 

wing-beat 

j  orce, 

gm. 

m.        per  sec 
(From  V. 

m./s. 
Bjerknes) 

vortex* 

gra. 

FjW 

Stork 

3500 

0-91              2 

5-7 

1-5 

1480 

2:5 

Gull 

1000 

0-60             3 

5-7 

1-0 

640 

2:3 

Pigeon 

350 

0-30             6 

5-7 

0-5 

160 

1:2 

Sparrow 

30 

Oil            13 

4-5 

0-18 

13 

2:5 

Bee 

0-07 

0-01          200 

6-3 

002 

0-2 

3i:l 

Fly 

001 

0-007        190 

4-2 

001 

004 

4:1 

*  Conjectural. 

A  bird  may  exert  a  force  at  each  stroke  of  its  wing  equal  to 
one-half,  let  us  say  for  safety  one-quarter,  of  its  own  weight,  more 
or  less ;  but  a  bee  or  a  fly  does  twice  or  thrice  the  equivalent  of  its 
own  weight,  at  a  low  estimate.  If  stork,  gull  or  pigeon  can  thus 
carry  only  one-fifth,  one-third,  one- quarter  of  their  weight  by  the 
beating  of  their  wings,  it  follows  that  all  the  rest  must  be  borne  by 
sailing-flight  between  the  wing-beats.  But  an  insect's  wings  lift  it 
easily  and  with  something  to  spare;  hence  saihng-flight,  and  with 
it  the  whole  principle  of  necessary  speed,  does  not  concern  the  lesser 
insects,  nor  the  smallest  birds,  at  all;  for  a  humming-bird  can 
"stand  still"  in  the  air,  like  a  hover-fly,  and  dart  backwards  as 
well  as  forwards,  if  it  please. 

There  is  a  little  group  of  Fairy-flies  (Mymaridae),  far  below  the 
size  of  any  small  famiUar  insects;  their  eggs  are  laid  and  larvae 
reared  within  the  tiny  eggs  of  larger  insects ;  their  bodies  may  be  no 
more  than  J  mm.  long,  and  their  outspread  wings  2  mm.  from  tip 
to  tip  (Fig.  2).  It  is  a  pecuharity  of  some  of  these  that  their  Httle 
wings  are  made  of  a  few  hairs  or  bristles,  instead  of  the  continuous 
membrane  of  a  wing.  How  these  act  on  the  minute  quantity  of  air 
involved  we  can  only  conjecture.  It  would  seem  that  that  small 
quantity  reacts  as  a  viscous  fluid  to  the  beat  of  the  wing ;  but  there 
are  doubtless  other  unobserved  anomahes  in  the  mechanism  and 
the  mode  of  flight  of  these  pigmy  creaturesf . 

The  ostrich  has  apparently  reached  a  magnitude,  and  the  moa 
certainly  did  so,  at  which  flight  by  muscular  action,  according  to 

t  It  is  obvious  that  in  a  still  smaller  order  of  magnitude  tfie  Brownian  movement 
would  suffice  to  make^  flight  impossible. 


48 


ON  MAGNITUDE 


[CH. 


the  normal  anatomy  of  a  bird,  becomes  physiologically  impossible. 
The  same  reasoning  applies  to  the  case  of  man.  It  would  be  very 
difficult,  and  probably  absolutely  impossible,  for  a  bird  to  flap  its 
way  through  the  air  were  it  of  the  bigness  of  a  man;  but  Borelli, 
in  discussing  the  matter,  laid  even  greater  stress  on  the  fact  that 
a  man's  pectoral  muscles  are  so  much  less  in  proportion  than  those 
of  a  bird,  that  however  we  might  fit  ourselves  out  with  wings,  we 
could  never  expect  to  flap  them  by  any  power  of  our  own  weak 
muscles.  Borelli  had  learned  this  lesson  thoroughly,  and  in  one  of 
his  chapters  he  deals  with  the  proposition :  Est  impossibile  ut  homines 
jyropriis  viribus  artificiose  volare  possinV^,     But  gliding  flight,  where 


a  '        b 

2.   Fairy-flies  (Mymaridae) :  after  F.  Enock.    x  20. 


wind-force  and  gravitational  energy  take  the  place  of  muscular 
power,  is  another  story,  and  its  limitations  are  of  another  kind. 
Nature  has  many  modes  and  mechanisms  of  flight,  in  birds  of  one 
kind  and  another,  in  bats  and  beetles,  butterflies,  dragonflies  and 
what  not ;  and  gliding  seems  to  be  the  common  way  of  birds,  and 
the  flapping  flight  {remigio  alarum)  of  sparrow  and  of  crow  to  be 
the  exception  rather  than  the  rule.  But  it  were  truer  to  say  that 
gliding  and  soaring,  by  which  energy  is  captured  from  the  wind,  are 
modes  of  flight  little  needed  by  the  small  birds,  but  more  and  more 
essential  to  the  large.  BorelH  had  proved  so  convincingly  that 
we  could  never  hope  to  fly  propriis  viribus,  that  all  through  the 
eighteenth  century  men  tried  no  more  to  fly  at  all.  It  was  in  trying 
to  glide  that  the  pioneers  of  aviation,  Cayley,  Wenham  and  Mouillard, 

*  Giovanni  Alfonso  Borelli,  De  Motu  Animalium,  i,  Prop,  cciv,  p.  243,  edit. 
1685.  The  part  on  The  Flight  of  Birds  is  issued  by  the  Royal  Aeronautical  Society 
as  No.  6  of  its  Aeronautical  Classics. 


II]  OF  GLIDING  FLIGHT  49 

Langley,  Lilienthal  and  the  Wrights — all  careful  students  of  birds — 
renewed  the  attempt*;  and  only  after  the  Wrights  had  learned  to 
ghde  did  they  seek  to  add  power  to  their  glider.  Flight,  as  the 
Wrights  declared,  is  a  matter  of  practice  and  of  skill,  and  skill  in 
gliding  has  now  reached  a  point  which  more  than  justifies  all 
Leonardo  da  Vinci's  attempts  to  fly.  Birds  shew  infinite  skill  and 
instinctive  knowledge  in  the  use  they  make  of  the  horizontal  accelera- 
tion of  the  wind,  and  the  advantage  they  take  of  ascending  currents 
in  the  air.  Over  the  hot  sands  of  the  Sahara,  where  every  here 
and  there  hot  air  is  going  up  and  cooler  coming  down,  birds  keep 
as  best  they  can  to  the  one,  or  ghde  quickly  through  the  other; 
so  we  may  watch  a  big  dragonfly  planing  slowly  down  a  few  feet 
above  the  heated  soil,  and  only  every  five  minutes  or  so  regaining 
height  with  a  vigorous  stroke  of  his  wings.  The  albatross  uses  the 
upward  current  on  the  lee-side  of  a  great  ocean- wave ;  so,  on  a  lesser 
scale,  does  the  flying- fish;  and  the  seagull  flies  in  curves,  taking 
every  advantage  of  the  varying  wind- velocities  at  different  levels 
over  the  sea.  An  Indian  vulture  flaps  his  way  up  for  a  few  laborious 
yards,  then  catching  an  upward  current  soars  in  easy  spirals  to 
2000  feet ;  here  he  may  stay,  effortless,  all  day  long,  and  come  down 
at  sunset.  Nor  is  the  modern  sail-plane  much  less  efiicient  than  a 
soaring  bird ;  for  a  skilful  pilot  in  the  tropics  should  be  able  to  roam 
all  day  long  at  willf . 

A  bird's  sensitiveness  to  air-pressure  is  indicated  in  other  ways 
besides.  Heavy  birds,  hke  duck  and  partridge,  fly  low  and  ap- 
parently take  advantage  of  air-pressure  reflected  from  the  ground. 
Water-hen  and  dipper  follow  the  windings  of  the  stream  as  they  fly 
up  or  down;  a  bee-hne  would  give  them  a  shorter  course,  but  not 
so  smooth  a  journey.  Some  small  birds — wagtails,  woodpeckers  and 
a  few  others — fly,  so  to  speak,  by  leaps  and  bounds ;  they  fly  briskly 

*  Sir  George  Cayley  (1774-1857),  father  of  British  aeronautics,  was  the  first  to 
perceive  the  capabilities  of  rigid  planes,  and  to  experiment  on  gliding  flight.  He 
anticipated  all  the  essential  principles  of  the  modern  aeroplane,  and  his  first  paper 
"On  Aerial  Navigation"  appeared  in  Nicholson's  Journal  for  November  1809. 
F.  H.  Wenham  (1824-1908)  studied  the  flight  of  birds  and  estimated  the  necessary 
proportion  of  surface  to  weight  and  speed;  he  held  that  "the  whole  secret  of 
success  in  flight  depends  upon  a  proper  concave  form  of  the  supporting  surface." 
See  his  paper  "On  Aerial  Locomotion"  in  the  Report  of  the  Aeronautical  Society 
1866. 

t  Sir  Gilbert  Walker,  in  Nature,  Oct.  2,  1937. 


50  ON  MAGNITUDE  [ch. 

for  a  few  moments,  then  close  their  wings  and  shoot  along*.  The 
flying-fishes  do  much  the  same,  save  that  they  keep  their  wings 
outspread.  The  best  of  them  "taxi"  along  with  only  their  tails  in 
the  water,  the  tail  vibrating  with  great  rapidity,  and  the  speed 
attained  lasts  the  fish  on  its  long  glide  through  the  airf. 

Flying  may  have  begun,  as  in  Man's  case  it  did,  with  short  spells 
of  gUding  flight,  helped  by  gravity,  and  far  short  of  sustained  or 
continuous  locomotion.  The  short  wings  and  long  tail  of  Archae- 
opteryx  would  be  efficient  as  a  slow-speed  ghder;  and  we  may  still 
see  a  Touraco  glide  down  from  his  perch  looking  not  much  unlike 
Archaeopteryx  in  the  proportions  of  his  wings  and  tail.  The  small 
bodies,  scanty  muscles  and  narrow  but  vastly  elongated  wings  of 
a  Pterodactyl  go  far  beyond  the  hmits  of  mechanical  efficiency  for 
ordinary  flapping  flight;  but  for  ghding  they  approach  perfection  J. 
Sooner  or  later  Nature  does  everything  which  is  physically  possible ; 
and  to  glide  with  skill  and  safety  through  the  air  is  a  possibihty 
which  she  did  not  overlook. 

Apart  from  all  differences  in  the  action  of  the  limbs — apart  from 
differences  in  mechanical  construction  or  in  the  manner  in  which 
the  mechanism  is  used — we  have  now  arrived  at  a  curiously  simple 
and  uniform  result.  For  in  all  the  three  forms  of  locomotion  which 
we  have  attempted  to  study,  alike  in  swimming  and  in  walking,  and 
even  in  the  more,  complex  problem  of  flight,  the  general  result, 
obtained  under  very  different  conditions  and  arrived  at  by  different 
modes  of  reasoning,  shews  in  every  case  that  speed  tends  to  vary  as 
the  square  root  of  the  linear  dimensions  of  the  animal. 

While  the  rate  of  progress  tends  to  increase  slowly  with  increasing 
size  (according  to  Froude's  law),  and  the  rhythm  or  pendulum-rate 
of  the  limbs  to  increase  rapidly  with  decreasing  size  (according  to 
Galileo's   law),    some   such   increase   of  velocity   with   decreasing 

*  Why  large  birds  cannot  do  the  same  is  discussed  by  Lanchester,  op.  cit. 
Appendix  iv., 

t  Cf.  Carl  L.  Hubbs,  On  the  flight  of. .  .the  Cypselurinae,  and  remarks  on  the 
evolution  of  the  flight  of  fishes.  Papers  of  the  Michigan  Acad,  of  Sci.  xvii,  pp.  575- 
611,  1933.  See  also  E.  H.  Hankin,  P.Z.S.  1920,  pp.  467-474;  and  C.  M.  Breeder, 
On  the  structural  specialisation  of  flying  fishes  from  the  standpoint  of  aero- 
dynamics, Copeia,  1930,  pp.  114-121. 

X  The  old  conjecture  that  their  flight  was  helped  or  rendered  possible  by  a  denser 
atmosphere  than  ours  is  thus  no  longer  called  for. 


II]  OF  THE  FORCE  OF  GRAVITY  51 

magnitude  is  true  of  all  the  rhythmic  actions  of  the  body,  though 
for  reasons  not  always  easy  to  explain.  The  elephant's  heart  beats 
slower  than  ours*,  the  dog's  quicker;  the  rabbit's  goes  pit-a-pat; 
the  mouse's  and  the  sparrow's  are  too  quick  to  count.  But  the  very 
"rate  of  Hving"  (measured  by  the  0  consumed  and  COg  produced) 
slows  down  as  size  increases;  and  a  rat  lives  so  much  faster  than 
a  man  that  the  years  of  its  life  are  three,  instead  of  threescore  and 
ten. 

From  all  the  foregoing  discussion  we  learn  that,  as  Crookes  once 
upon  a  time  remarked  I,  the  forms  as  well  as  the  actions  of  our 
bodies  are  entirely  conditioned  (save  for  certain  exceptions  in  the 
case  of  aquatic  animals)  by  the  strength  of  gravity  upon  this  globe; 
or,  as  Sir  Charles  Bell  had  put  it  some  sixty  years  before,  the  very 
animals  which  move  upon  the  surface  of  the  earth  are  proportioned 
to  its  magnitude.  Were  the  force  of  gravity  to  be  doubled  our 
bipedal  form  would  be  a  failure,  and  the  majority  of  terrestrial 
animals  would  resemble  short-legged  saurians,  or  else  serpents. 
Birds  and  insects  would  suffer  Ukewise,  though  with  some  com- 
pensation in  the  increased  density  of  the  air.  On  the  other  hand, 
if  gravity  were  halved,  we  should  get  a  lighter,  slenderer,  more  active 
type,  needing  less  energy,  less  heat,  less  heart,  less  lungs,  less  blood. 
Gravity  not  only  controls  the  actions  but  also  influences  the  forms 
of  all  save  the  least  of  organisms.  The  tree  under  its  burden  of 
leaves  or  fruit  has  changed  its  every  curve  and  outline  since  its 
boughs  were  bare,  and  a  mantle  of  snow  will  alter  its  configuration 
again.  Sagging  wrinkles,  hanging  breasts  and  many  another  sign 
of  age  are  part  of  gravitation's  slow  relentless  handiwork. 

There  are  other  physical  factors  besides  gravity  which  help  to 
limit  the  size  to  which  an  animal  may  grow  and  to  define  the  con- 
ditions under  which  it  may  Hve.  The  small  insects  skating  on  a 
pool  have  their  movements  controlled  and  their  freedom  hmited  by 
the  surface-tension  between  water  and  air,  and  the  measure  of  that 
tension  determines  the  magnitude  which  they  may  attain.  A  man 
coming  wet  from  his  bath  carries  a  few  ounces  of  water,  and  is 
perhaps  1  per  cent,  heavier  than  before;  but  a  wet  fly  weighs  twice 
as  much  as  a  dry  one,  and  becomes  a  helpless  thing.  *  A  small 

*  Say  28  to  30  beats  to  the  minute. 

t  Proc.  Psychical  Soc.  xn,  p.  338-355,  1^97. 


52  ON  MAGNITUDE  [ch. 

insect  finds  itself  imprisoned  in  a  drop  of  water,  and  a  fly  with 
two  feet  in  one  drop  finds  it  hard  to  extricate  them. 

The  mechanical  construction  of  insect  or  crustacean  is  highly 
efficient  up  to  a  certain  size,  but  even  crab  and  lobster  never  exceed 
certain  moderate  dimensions,  perfect  within  these  narrow  bounds  as 
their  construction  seems  to  be.  Their  body  lies  within  a  hollow 
shell,  the  stresses  within  which  increase  much  faster  than  the  mere 
scale  of  size ;  every  hollow  structure,  every  dome  or  cyhnder,  grows 
weaker  as  it  grows  larger,  and  a  tin  canister  is  easy  to  make  but  a 
great  boiler  is  a  complicated  affair.  The  boiler  has  to  be  strengthened 
by  "stiffening  rings"  or  ridges,  and  so  has  the  lobster's  shell;  but 
there  is  a  limit  even  to  this  method  of  counteracting  the  weakening 
effect  of  size.  An  ordinary  girder-bridge  may  be  made»efficient  up 
to  a  span  of  200  feet  or  so ;  but  it  is  physically  incapable  of  spanning 
the  Firth  of  Forth.  The  great  Japanese  spider-crab,  Macrocheira, 
has  a  span  of  some  12  feet  across;  but  Nature  meets  the  difficulty 
and  solves  the  problem  by  keeping  the  body  small,  and  building  up 
the  long  and  slender  legs  out  of  short  lengths  of  narrow  tubes. 
A  hollow  shell  is  admirable  for  small  animals,  but  Nature  does  not 
and  cannot  make  use  of  it  for  the  large. 

In  the  case  of  insects,  other  causes  help  to  keep  them  of  small 
dimensions.  In  their  peculiar  respiratory  system  blood  does  not 
carry  oxygen  to  the  tissues,  but  innumerable  fine  tubules  or  tracheae 
lead  air  into  the  interstices  of  the  body.  If  we  imagine  them  growing 
even  to  the  size  of  crab  or  lobster,  a  vast  complication  of  tracheal 
tubules  would  be  necessary,  within  which  friction  would  increase 
and  diffusion  be  retarded,  and  which  would  soon  be  an  inefficient 
and  inappropriate  mechanism. 

The  vibration  of  vocal  chords  and  auditory  drums  has  this  in 
common  with  the  pendulum-hke  motion  of  a  hmb  that  its  rate 
also  tends  to  vary  inversely  as  the  square  root  of  the  linear  dimen- 
sions. We  know  by  common  experience  of  fiddle,  drum  or  organ, 
that  pitch  rises,  or  the  frequency  of  vibration  increases,  as  the 
dimensions  of  pipe  or  membrane  or  string  diminish;  and  in  like 
manner  we  expect  to  hear  a  bass  note  from  the  great  beasts  and  a 
piping  treble  from  the  small.  The  rate  of  vibration  {N)  of  a  stretched 
string  depends  on  its  tension  and  its  density;  these  beins^  equal,  it 
varies  inversely  as  its  own  length  and  as  its  diameter,     i^or  similar 


II]  OF  EYES  AND  EARS  53 

strings,  N  oc  1//^,  and  for  a  circular  membrane,  of  radius  r  and 
thickness  e,  N  oc  ll(r^  Ve). 

But  the  deHcate  drums  or  tympana  of  various  animals  seem  to 
vary  much  less  in  thickness  than  in  diameter,  and  we  may  be  content 
to  write,  once  more,  N  oc  l/r^. 

Suppose  one  animal  to  be  fifty  times  less  than  another,  vocal 
chords  and  all:  the  one's  voice  will  be  pitched  2500  times  as  many 
beats,  or  some  ten  or  eleven  octaves,  above  the  other's;  and  the 
same  comparison,  or  the  same  contrast,  will  apply  to  the  tympanic 
membranes  by  which  the  vibrations  are  received.  But  our  own 
perception  of  musical  notes  only  reaches  to  4000  vibrations  per 
second,  or  thereby;  a  squeaking  mouse  or  bat  is  heard  by  few,  and 
to  vibrations  of  10,000  per  second  we  are  all  of  us  stone-deaf. 
Structure  apart,  mere  size  is  enough  to  give  the  lesser  birds  and 
beasts  a  music  quite  different  to  our  own:  the  humming-bird,  for 
aught  we  know,  may  be  singing  all  day  long.  A  minute  insect  may 
utter  and  receive  vibrations  of  prodigious  rapidity;  even  its  little 
wings  may  beat  hundreds  of  times  a  second*.  Far  more  things 
happen  to  it  in  a  second  than  to  us ;  a  thousandth  part  of  a  second 
is  no  longer  neghgible,  and  time  itself  seems  to  run  a  different  course 
to  ours. 

The  eye  and  its  retinal  elements  have  ranges  of  magnitude  and 
Hmitations  of  magnitude  of  their  own.  A  big  dog's  eye  is  hardly 
bigger  than  a  little  dog's;  a  squirrel's  is  much  larger,  propor- 
tionately, than  an  elephant's;  and  a  robin's  is  but  little  less  than 
a  pigeon's  or  a  crow's.  For  the  rods  and  cones  do  not  vary  with 
the  size  of  the  animal,  but  have  their  dimensions  optically  limited 
by  the  interference-patterns  of  the  waves  of  light,  which  set  bounds 
to  the  production  of  clear  retinal  images.  True,  the  larger  animal 
may  want  a  larger  field  of  view ;  but  this  makeg  little  difference,  for 
but  a  small  area  of  the  retina  is  ever  needed  or  used.  The  eye,  in 
short,  can  never  be  very  small  and  need  never  be  very  big;  it  has 
its  own  conditions  and  limitations  apart  from  the  size  of  the  animal. 
But  the  insect's  eye  tells  another  story.  If  a  fly  had  an  eye  like 
ours,  the  pupil  would  be  so  small  that  diffraction  would  render  a 
clear  image  impossible.     The  only  alternative  is  to  unite  a  number 

*  The  wing-beats  are  said  to  be  as  follows:  dragonfly  28  per  sec,  bee  190, 
housefly  330;   cf.  Erhard,  Verh.  d.  d.  zool.  Gesellsch.  1913,  p.  206. 


54  ON  MAGNITUDE  [ch. 

of  small  and  optically  isolated  simple  eyes  into  a  compound  eye, 
and  in  the  insect  Nature  adopts  this  alternative  possibiHty*. 

Our  range  of  vision  is  limited  to  a  bare  octave  of  "luminous" 
waves,  which  is  a  considerable  part  of  the  whole  range  of  Hght-heat 
rays  emitted  by  the  sun;  the  sun's  rays  extend  into  the  ultra-violet 
for  another  half-octave  or  more,  but  the  rays  to  which  our  eyes  are 
sensitive  are  just  those  which  pass  with  the  least  absorption  through 
a  watery  medium.  Some  ancient  vertebrate  may  have  learned  to 
see  in  an  ocean  which  let  a  certain  part  of  the  sun's  whole  radiation 
through,  which  part  is  our  part  still ;  or  perhaps  the  watery  media 
of  the  eye  itself  account  sufficiently  for  the  selective  filtration.  In 
either  case,  the  dimensions  of  the  retinal  elements  are  so  closely 
related  to  the  wave-lengths  of  light  (or  to  their  interference  patterns) 
that  we  have  good  reason  to  look  upon  the  retina  as  perfect  of  its 
kind,  within  the  hmits  which  the  properties  of  hght  itself  impose; 
and  this  perfection  is  further  illustrated  by  the  fact  that  a  few 
light-quanta,  perhaps  a  single  one,  suffice  to  produce  a  sensation  "j". 
The  hard  eyes  of  insects  are  sensitive  over  a  wider  range.  The  bee 
has  two  visual  optima,  one  coincident  with  our  own,  the  other  and 
principal  one  high  up  in  the  ultra-violet  J.  And  with  the  latt^er  the 
bee  is  able  to  see  that  ultra-violet  which  is  so  well  reflected  by  many 
flowers  that  flower-photographs  have  been  taken  through  a  filter 
which  passes  these  but  transmits  no  other  rays§. 

When  we  talk  of  hght,  and  of  magnitudes  whose  order  is  that  of 
a  wave-length  of  hght,  the  subtle  phenomenon  of  colour  is  near  at 
hand.  The  hues  of  hving  things  are  due  to  sundry  causes;  where 
they  come  from  chemical  pigmentation  they  are  outside  our  theme, 
but  oftentimes  there  is  no  pigment  at  all,  save  perhaps  as  a  screen 
or  background,  and  the  tints  are  those  proper  to  a  scale  of  wave- 
lengths or  range  of  magnitude.  In  birds  these  "optical  colours" 
are  of  two  chief  kinds.     One  kind  include  certain  vivid  blues,  the 

*  Cf.  C.  J.  van  der  Horst,  The  optics  of  the  insect  eye,  Acta  Zoolog.  1933, 
p.  108. 

t  Cf.  Niels  Bohr,  in  Nature,  April  1,  1933,  p.  457.  Also  J.  Joly,  Proc.  R.S.  (B), 
xcn,  p.  222,  1921. 

f  L.  M.  Bertholf,  Reactions  of  the  honey-bee  to  light,  Journ.  of  Agric.  Res. 
XLm,  p.  379;   xliv,  p.  763,  1931. 

§  A.  Kuhn,  Ueber  den  Farbensinn  der  Bienen,  Ztschr.  d.  vergl.  Physiol,  v, 
pp.  762-800,  1927;  cf.  F.  K.  Richtmeyer,  Reflection  of  ultra-violet  by  flowers, 
J  num.  Optical  Soc.  Amer.  vii,  pp.  151-168,  1923;   etc. 


II]  OF  LIGHT  AND  COLOUR  55 

blue  of  a  blue  jay,  an  Indian  roller  or  a  macaw;  to  the  other  belong 
the  iridescent  hues  of  mother-of-pearl,  of  the  humming-bird,  the 
peacock  and  the  dove:  for  the  dove's  grey  breast  shews  many 
colours  yet  contains  but  one — colores  inesse  plures  nee  esse  plus  uno, 
as  Cicero  said.  The  jay's  blue  feather  shews  a  layer  of  enamel-like 
cells  beneath  a  thin  horny  cuticle,  and  the  cell-walls  are  spongy 
with  innumerable  tiny  air-filled  pores.  These  are  about  0-3 /it  in 
diameter,  in  some  birds  even  a  little  less,  and  so  are  not  far  from 
the  hmits  of  microscopic  vision.  A  deeper  layer  carries  dark-brown 
pigment,  but  there  is  no  blue  pigment  at  all;  if  the  feather  be  dipped 
in  a  fluid  of  refractive  index  equal  to  its  own,  the  blue  utterly 
disappears,  to  reappear  when  the  feather  dries.  This  blue  is  like 
the  colour  of  the  sky;  it  is  ''Tyndall's  blue,"  such  as  is  displayed 
by  turbid  media,  cloudy  with  dust-motes  or  tiny  bubbles  of  a  size 
comparable  to  the  wave-lengths  of  the  blue  end  of  the  spectrum. 
The  longer  waves  of  red  or  yellow  pass  through,  the  shorter  violet 
rays  are  reflected  or  scattered;  the  intensity  of  the  blue  depends 
on  the  size  and  concentration  of  the  particles,  while  the  dark  pigment- 
screen  enhances  the  effect. 

Rainbow  hues  are  more  subtle  and  more  complicated ;  but  in  the 
peacock  and  the  humming-bird  we  know  for  certain*  that  the 
colours  are  those  of  Newton's  rings,  and  are  produced  by  thin  plates 
or  films  covering  the  barbules  of  the  feather.  The  colours  are  such 
as  are  shewn  by  films  about  J  [x  thick,  more  or  less ;  they  change 
towards  the  blue  end  of  the  spectrum  as  thei  hght  falls  more  and 
more  obliquely;  or  towards  the  red  end  if  you  soak  the  feather 
and  cause  the  thin  plates  to  swell.  The  barbules  of  the  peacock's 
feather  are  broad  and  flat,  smooth  and  shiny,  and  their  cuticular 
layer  sphts  into  three  very  thin  transparent  films,  hardly  more  than 
1  jjL  thick,  all  three  together.  The  gorgeous  tints  of  the  humming- 
birds have  had  their  places  in  Newton's  scale  defined,  and  the 
changes  which  they  exhibit  at  varying  incidence  have  been  predicted 

*  Rayleigh,  Phil.  Mag.  (6),  xxxvii,  p.  98,  1919.  For  a  review  of  the  whole 
subject,  and  a  discussion  of  its  many  difficulties,  see  H.  Onslow,  On  a  periodic 
structure  in  many  insect  scales,  etc.,  Phil.  Trans.  (B),  ccxi,  pp.  1-74,  1921; 
also  C.  W.  Mason,  Journ.  Physic.  Chemistry,  xxvii,  xxx,  xxxi,  1923-25-27; 
F.  Suffert,  Zeitschr.  f.  Morph.  u.  Oekol.  d.  Tiere,  i,  pp.  171-306,  1924  (scales  of 
butterflies);  also  B.  Reusch  and  Th.  Elsasser  in  Journ.  f.  Ornithologie,  lxxiti, 
1925;   etc. 


56  ON  MAGNITUDE  [ch. 

and  explained.  The  thickness  of  each  film  lies  on  the  very  limit  of 
microscopic  vision,  and  the  least  change  or  irregularity  in  this 
minute  dimension  would  throw  the  whole  display  of  colour  out  of 
gear.  No  phenomenon  of  organic  magnitude  is  more  striking  than 
this  constancy  of  size;  none  more  remarkable  than  that  these  fine 
lamellae  should  have  their  tenuity  so  sharply  defined,  so  uniform 
in  feather  after  feather,  so  identical  in  all  the  individuals  of  a  species, 
so  constant  from  one  generation  to  another. 

A  simpler  phenomenon,  and  one  which  is  visible  throughout  the 
whole  field  of  morphology,  is  the  tendency  (referable  doubtless  in 
each  case  to  some  definite  physical  cause)  for  mere  bodily  surface 
to  keep  pace  with  volume,  through  some  alteration  of  its  form.  The 
development  of  villi  on  the  lining  of  the  intestine  (which  increase 
its  surface  much  as  we  enlarge  the  effective  surface  of  a  bath-towel), 
the  various  valvular  folds  of  the  intestinal  lining,  including  the 
remarkable  "spiral  valve"  of  the  shark's  gut,  the  lobulation  of  the 
kidney  in  large  animals*,  the  vast  increase  of  respiratory  surface  in 
the  air-sacs  and  alveoli  of  the  lung,  the  development  of  gills  in  the 
larger  Crustacea  and  worms  though  the  general  surface  of  the  body 
suffices  for  respiration  in  the  smaller  species — all  these  and  many 
more  are  cases  in  which  a  more  or  less  constant  ratio  tends  to  be 
maintained  between  mass  and  surface,  which  ratio  would  have  been 
more  and  more  departed  from  with  increasing  size,  had  it  not  been 
for  such  alteration  of  surface-form  f.  A  leafy  wood,  a  grassy  sward, 
a  piece  of  sponge,  a  reef  of  coral,  are  all  instances  of  a  hke  pheno- 
menon. In  fact,  a  deal  of  evolution  is  involved  in  keeping  due 
balance  between  surface  and  mass  as  growth  goes  on. 

In  the  case  of  very  small  animals,  and  of  individual  cells,  the 
principle  becomes  especially  important,  in  consequence  of  the 
molecular  forces  whose  resultant  action  is  limited  to  the  superficial 
layer.  In  the  cases  just  mentioned,  action  is  facilitated  by  increase 
of  surface :  diffusion,  for  instance,  of  nutrient  liquids  or  respiratory 
gases  is  rendered  more  rapid  by  the  greater  area  of  surface;    but 

*  Cf.  R.  Anthony,  C.R.  clxix,  p.  1174,  1919,  etc.  Cf.  also  A.  Putter,  Studien 
uber  physiologische  Ahnlichkeit,  Pfluger's  Archiv,  clxviii,  pp.  209-246,  1917. 

t  For  various  calculations  of  the  increase  of  surface  due  to  histological  and 
anatomical  subdivision,  see  E.  Babak,  Ueber  die  Oberflachenentwickelung  bei 
Organismen,  Biol.  Centralbl.  xxx,  pp.  225-239,  257-267,  1910. 


II]  OF  SURFACE  AND  VOLUME  57 

there  are  other  cases  in  which  the  ratio  of  surface  to  mass  may 
change  the  whole  condition  of  the  system.  Iron  rusts  when  exposed 
to  moist  air,  but  it  rusts  ever  so  much  faster,  and  is  soon  eaten  away, 
if  the  iron  be  first  reduced  to  a  heap  of  small  filings ;  this  is  a  mere 
difference  of  degree.  But  the  spherical  surface  of  the  rain-drop 
and  the  spherical  surface  of  the  ocean  (though  both  happen  to  be 
ahke  in  mathematical  form)  are  two  totally  different  phenomena, 
the  one  due  to  surface-energy,  and  the  other  to  that  form  of  mass- 
energy  which  we  ascribe  to  gravity.  The  contrast  is  still  more 
clearly  seen  in  the  case  of  waves:  for  the  little  ripple,  whose  form 
and  manner  of  propagation  are  governed  by  surface-tension,  is 
found  to  travel  with  a  velocity  which  is  inversely  as  the  square 
root  of  its  length;  while  the  ordinary  big  waves,  controlled  by 
gravitation,  have  a  velocity  directly  proportional  to  the  square  root 
of  their  wave-length.  In  hke  manner  we  shall  find  that  the  form 
of  all  very  small  organisms  is  independent  of  gravity,  and  largely 
if  not  mainly  due  to  the  force  of  surface-tension:  either  as  the 
direct  result  of  the  continued  action  of  surface-tension  on  the 
semi-fluid  body,  or  else  as  the  result  of  its  action  at  a  prior  stage 
of  development,  in  bringing  about  a  form  which  subsequent  chemical 
changes  have  rendered  rigid  and  lasting.  In  either  case,  we  shall 
find  a  great  tendency  in  small  organisms  to  assume  either  the 
spherical  form  or  other  simple  forms  related  to  ordinary  inanimate 
surface-tension  phenomena,  which  forms  do  not  recur  in  the 
external  morphology  of  large  animals. 

Now  this  is  a  very  important  matter,  and  is  a  notable  illustration 
of  that  principle  of  simihtude  which  we  have  already  discussed  in 
regard  to  several  of  its  manifestations.  We  are  coming  to  a  con- 
clusion which  will  affect  the  whole  course  of  our  argument  throughout 
this  book,  namely  that  there  is  an  essential  difference  in  kind 
between  the  phenomena  of  form  in  the  larger  and  the  smaller 
organisms.  I  have  called  this  book  a  study  of  Growth  and  Fonn, 
because  in  the  most  familiar  illustrations  of  organic  form,  as  in  our 
own  bodies  for  example,  these  two  factors  are  inseparably  asso- 
ciated, and  because  we  are  here  justified  in  thinking  of  form  as  the 
direct  resultant  and  consequence  of  growth:  of  growth,  whose 
varying  rate  in  one  direction  or  another  has  produced,  by  its  gradual 
and  unequal  increments,  the  successive  stages  of  development  and 


58  ON  MAGNITUDE  [ch. 

the  final  configuration  of  the  whole  material  structure.  But  it  is 
by  no  means  true  that  form  and  growth  are  in  this  direct  and  simple 
fashion  correlative  or  complementary  in  the  case  of  minute  portions 
of  living  matter.  For  in  the  smaller  organisms,  and  in  the  indi- 
vidual cells  of  the  larger,  we  have  reached  an  order  of  magnitude 
in  which  the  intermolecular  forces  strive  under  favourable  conditions 
with,  and  at  length  altogether  outweigh,  the  force  of  gravity,  and 
also  those  other  forces  leading  to  movements  of  convection  which 
are  the  prevailing  factors  in  the  larger  material  aggregate. 

However,  we  shall  require  to  deal  more  fully  with  this  matter  in 
our  discussion  of  the  rate  of  growth,  and  we  may  leave  it  mean- 
while, in  order  to  deal  with  other  matters  more  or  less  directly 
concerned  with  the  magnitude  of  the  cell. 

The  hving  cell  is  a  very  complex  field  of  energy,  and  of  energy 
of  many  kinds,  of  which  surface-energy  is  not  the  least.  Now  the 
whole  surface-energy  of  the  cell  is  by  no  means  restricted  to  its 
outer  surface;  for  the  cell  is  a  very  heterogeneous  structure,  and  all 
its  protoplasmic  alveoli  and  other  visible  (as  well  as  invisible)  hetero- 
geneities make  up  a  great  system  of  internal  surfaces,  at  every  part 
of  which  one  "phase"  comes  in  contact  with  another  "phase,"  and 
surface-energy  is  manifested  accordingly.  But  still,  the  external 
surface  is  a  definite  portion  of  the  system,  with  a  definite  "phase" 
of  its  own,  and  however  little  we  may  know  of  the  distribution  of 
the  total  energy  of  the  system,  it  is  at  least  plain  that  the  conditions 
which  favour  equihbrium  will  be  greatly  altered  by  the  changed 
ratio  of  external  surface  to  mass  which  a  mere  change  of  magnitude 
produces  in  the  cell.  In  short,  the  phenomenon  of  division  of  the 
growing  cell,  however  it  be  brought  about,  will  be  precisely  what 
is  wanted  to  keep  fairly  constant  the  ratio  between  surface  and 
mass,  and  to  retain  or  restore  the  balance  between  surface-energy 
and  the  other  forces  of  the  system*.  But  when  a  germ-cell  divides 
or  "segments"  into  two,  it  does  not  increase  in  mass;  at  least  if 
there  be  some  shght  alleged  tendency  for  the  egg  to  increase  in 

*  Certain  cells  of  the  cucumber  were  found  to  divide  when  they  had  grown  to 
a  volume  half  as  large  again  as  that  of  the  "resting  cells."  Thus  the  volumes 
of  resting,  dividing  and  daughter  cells  were  as  1:1-5:  0-75;  and  their  surfaces, 
being  as  the  power  2/3  of  these  figures,  were,  roughly,  as  1:1-3:  0-8.  The  ratio 
of  SjV  was  then  as  1  :  0-9  :  1-1,  or  much  nearer  equality.  Cf.  F.  T.  Lewis,  Anat. 
Record,  xlvii,  pp.  59-99,  1930. 


II]  OF  THE  SIZE  OF  DROPS  59 

mass  or  volume  during  segmentation  it  is  very  slight  indeed, 
generally  imperceptible,  and  wholly  denied  by  some*.  The  growth 
or  development  of  the  egg  from  a  one-celled  stage  to  stages  of  two 
or  many  cells  is  thus  a  somewhat  pecuhar  kind  of  growth;  it  is 
growth  limited  to  change  of  form  and  increase  of  surface,  unaccom- 
panied by  growth  in  volume  or  in  mass.  In  the  case  of  a  soap-bubble, 
by  the  way,  if  it  divide  into  two  bubbles  the  volume  is  actually 
diminished,  while  the  surface-area  is  greatly  increased! ;  the  diminution 
being  due  to  a  cause  which  we  shall  have  to  study  later,  namely  to 
the  increased  pressure  due  to  the  greater  curvature  of  the  smaller 
bubbles. 

An  immediate  and  remarkable  result  of  the  principles  just 
described  is  a  tendency  on  the  part  of  all  cells,  according  to  their 
kind,  to  vary  but  little  about  a  certain  mean  size,  and  to  have  in 
fact  certain  absolute  limitations  of  magnitude.  The  diameter  of  a 
large  parenchymatous  cell  is  perhaps  tenfold  that  of  a  httle  one; 
but  the  tallest  phanerogams  are  ten  thousand  times  the  height  of 
the  least.  In  short,  Nature  has  her  materials  of  predeterminate 
dimensions,  and  keeps  to  the  same  bricks  whether  she  build  a  great 
house  or  a  small.  Even  ordinary  drops  tend  towards  a  certain 
fixed  size,  which  size  is  a  function  of  the  surface-tension,  and  may 
be  used  (as  Quincke  used  it)  as  a  measure  thereof.  In  a  shower  of 
rain  the  principle  is  curiously  illustrated,  as  Wilding  Roller  and 
V.  Bjerknes  tell  us.  The  drops  are  of  graded  sizes,  each  twice  as  big 
as  another,  beginning  with  the  minute  and  uniform  droplets  of  an 
impalpable  mist.  They  rotate  as  they  fall,  and  if  two  rotate  in 
contrary  directions  they  draw  together  and  presently  coalesce;  but 
this  only  happens  when  two  drops  are  faUing  side  by  side,  and  since 
the  rate  of  fall  depends  on  the  size  it  always  is  a  pair  of  coequal 
drops  which  so  meet,  approach  and  join  together.  A  supreme 
instance  of  constancy  or  quasi-constancy  of  size,  remote  from  but 
yet  analogous  to  the  size-hmitation  of  a  rain-drop  or  a  cell,  is  the 
fact  that  the  stars  of  heaven  (however  else  one  diifereth  from 
another),  and  even  the  nebulae  themselves,  are  all  wellnigh  co-equal 
in  mass.     Gravity  draws  matter  together,  condensing  it  into  a  world 

*  Though  the  entire  egg  is  not  increasing  in  mass,  that  is  not  to  say  that  its  living 
protoplasm  is  not  increasing  all  the  while  at  the  expense  of  the  reserve  material, 
t  Cf.  P.  G.  Tait,  Proc.  E.S.E.  v,  1866  and  vi,  1868. 


60  ON  MAGNITUDE  [ch. 

or  into  a  star;  but  ethereal  pressure  is  an  opponent  force  leading 
to  disruption,  negligible  on  the  small  scale  but  potent  on  the  large. 
High  up  in  the  scale  of  magnitude,  from  about  10^^  to  10^^  grams 
of  matter,  these  two  great  cosmic  forces  balance  one  another;  and 
all  the  magnitudes  of  all  the  stars  he  within  or  hard  by  these  narrow 
limits. 

In  the  hving  cell,  Sachs  pointed  out  (in  1895)  that  there  is  a 
tendency  for  each  nucleus  to  gather  around  itself  a  certain  definite 
amount  of  protoplasm*.  Drieschf,  a  httle  later,  found  it  possible, 
by  artificial  subdivision  of  the  egg,  to  rear  dwarf  sea-urchin  larvae, 
one-half,  one-quarter  or  even  one-eighth  of  their  usual  size;  which 
dwarf  larvae  were  composed  of  only  a  half,  a  quarter  or  an  eighth 
of  the  normal  number  of  cells.  These  observations  have  been  often 
repeated  and  amply  confirmed :  and  Loeb  found  the  sea-urchin  eggs 
capable  of  reduction  to  a  certain  size,  but  no  further. 

In  the  development  of  Crepidula  (an  American  "shpper-Hmpet," 
now  much  at  home  on  our  oyster-beds),  ConklinJ  has  succeeded  in 
rearing  dwarf  and  giant  individuals,  of  which  the  latter  may  be 
five-and-twenty  times  as  big  as  the  former.  But  the  individual 
cells,  of  skin,  gut,  liver,  muscle  and  other  tissues,  are  just  the  same 
size  in  one  as  in  the  other,  in  dwarf  and  in  giant  §.     In  like  manner 

*  Physiologische  Notizen  (9),  p.  425,  1895.  Cf.  Amelung,  Flora,  1893;  Stras- 
biirger,  Ueber  die  Wirkungssphare  der  Kerne  und  die  Zellgrosse,  Histol.  Beitr.  (5), 
pp.  95-129,  1893;  R.  Hertwig,  Ueber  Korrelation  von  Zell-  und  Kerngrosse 
(Kernplasraarelation),  Biol.  Centralbl.  xvm,  pp.  49-62,  108-119,  1903;  G.  Levi 
and  T.  Terni,  Le  variazioni  dell'  indice  plasmatico-nucleare  durante  1'  intercinesi, 
Arch.  Hal.  di  Anat.  x,  p.  545,  1911;  also  E.  le  Breton  and  G.  Schaeffer,  Variations 
hiuchimiqiies  du  rapport  nucleo-plasmatique,  Strasburg,  1923. 

t  Arch.f.  Entw.  Mech.  iv,  1898,  pp.  75,  247. 

X  E.  G.  Conklin,  Cell-size  and  nuclear  size,  Journ.  Exp.  Zool.  xii,  pp.  1-98, 
1912;  Body-size  and  cell-size,  Journ.  of  Morphol.  xxiii,  pp.  159-188,  1912.  Cf. 
M.  Popoff,  Ueber  die  Zellgrosse,  Arch.f.  Zellforschung,  iii,  1909. 

§  Thus  the  fibres  of  the  crystalline  lens  are  of  the  same  size  in  large  and  small 
dogs,  Rabl,  Z.f.  w.  Z.  lxvii,  1899.  Cf.  {int.  al.)  Pearson,  On  the  size  of  the  blood- 
corpuscles  in  Rana,  Biometrika,  vi,  p.  403,  1909.  Dr  Thomas  Young  caught  sight 
of  the  phenomenon  early  in  last  century:  "The  solid  particles  of  the  blood  do  not 
by  any  means  vary  in  magnitude  in  the  same  ratio  with  the  bulk  of  the  animal," 
Natural  Philosophy,  ed.  1845,  p.  466;  and  Leeuwenhoek  and  Stephen  Hales 
were  aware  of  it  nearly  two  hundred  years  before.  Leeuwenhoek  indeed  had 
a  very  good  idea  of  the  size  of  a  human  blood-corpuscle,  and  was  in  the  habit  of 
using  its  diameter — about  1/3000  of  an  inch — as  a  standard  of  comparison.  But 
though  the  blood-corpuscles  shew  no  relation  of  magnitude  to  the  size  of  the 
animal,  they  are  related  without  doubt  to  its  activity;    for  the  corpuscles  in  the 


OF  THE  SIZE  OF  CELLS 


61 


the  leaf-cells  are  found  to  be  of  the  same  size  in  an  ordinary  water- 
lily,  in  the  great  Victoria  regia,  and  in  the  still  hiiger  leaf,  nearly 
3  metres  long,  of  Euryale  ferox  in  Japan*.  Driesch  has  laid  par- 
ticular stress  upon  this  principle  of  a  "fixed  cell-size,"  which  has, 
however,  its  own  limitations  and  exceptions.  Among  these  excep- 
tions, or  apparent  exceptions,  are  the  giant  frond-like  cell  of  a 
Caulerpa  or  the  great  undivided  plasmodium  of  a  Myxomycete. 
The  flattening  of  the  one  and  the  branching  of  the  other  serve  (or 
help)  to  increase  the  ratio  of  surface  to  content,  the  nuclei  tend  to 
multiply,  and  streaming  currents  keep  the  interior  and  exterior  of 
the  mass  in  touch  with  one  another. 


j^ 


Rabbit 


Man  Dog 

Fig.  3.     Motor  ganglion-cells,  from  the  cervical  spinal  cord. 
From  Minot,  after  Irving  Hardesty. 

We  get  a  good  and  even  a  famiUar  illustration  of  the  principle 
of  size-hmitation  in  comparing  the  brain-cells  or  ganghon-cells, 
whether  of  the  lower  or  of  the  higher  animals  f .  In  Fig.  3  we  shew 
certain  identical  nerve-cells  from  various  mammals,  from  mouse  to 
elephant,  all  drawn  to  the  same  scale  of  magnification ;  and  we  see 
that  they  are  all  of  much  the  same  order  of  magnitude.  The  nerve- 
cell  of  the  elephant  is  about  twice  that  of  the  mouse  in  linear 

sluggish  Amphibia  are  much  the  largest  known  to  us,  while  the  smallest  are  found 
among  the  deer  and  other  agile  and  speedy  animals  (cf.  Gulliver,  P.Z.S.  1875, 
p.  474,  etc.).  This  correlation  is  explained  by  the  surface  condensation  or 
adsorption  of  oxygen  in  the  blood-corpuscles,  a  process  greatly  facilitated  and 
intensified  by  the  increase  of  surface  due  to  their  minuteness. 

*  Okada  and  Yomosuke,  in  Sci.  Rep.  Tohoku  Univ.  iii,  pp.  271-278,  1928. 

t  Cf.  P.  Enriques,  La  forma  eome  funzione  della  grandezza :  Ricerche  sui  gangli 
nervosi  degli  invertebrati,  Arch.  f.  Entw.  Mech.  xxv,  p.  655,  1907-8. 


62  ON  MAGNITUDE  [ch. 

dimensions,  and  therefore  about  eight  times  greater  in  volume  or 
in  mass.  But  making  due  allowance  for  difference  of  shape,  the 
linear  dimensions  of  the  elephant  are  to  those  of  the  mouse  as  not 
less  than  one  to  fifty ;  and  the  bulk  of  the  larger  animal  is  something 
like  125,000  times  that  of  the  less.  It  follows,  if  the  size  of  the 
nerve-cells  are  as  eight  to  one,  that,  in  corresponding  parts  of  the 
nervous  system,  there  are  more  than  15,000  times  as  many  individual 
cells  in  one  animal  as  in  the  other.  In  short  we  may  (with  Enriques) 
lay  it  down  as  a  general  law  that  among  animals,  large  or  small,  the 
gangUon-cells  vary  in  size  within  narrow  limits;  and  that,  amidst 
all  the  great  variety  of  structure  observed  in  the  nervous  system 
of  different  classes  of  animals,  it  is  always  found  that  the  smaller 
species  have  simpler  gangha  than  the  larger,  that  is  to  say  ganglia 
containing  a  smaller  number  of  cellular  elements*.  The  bepiing  of 
such  facts  as  this  upon  the  cell-theory  in  general  is  not  to  be  dis- 
regarded; and  the  warning  is  especially  clear  against  exaggerated 
attempts  to  correlate  physiological  processes  with  the  visible 
mechanism  of  associated  cells,  rather  than  with  the  system  of 
energies,  or  the  field  of  force,  which  is  associated  with  them.  For 
the  life  of  the  body  is  more  than  the  swm  of  the  properties  of  the 
cells  of  which  it  is  composed:  as  Goethe  said,  "Das  Lebendige  ist 
zwar  in  Elemente  zerlegt,*  aber  man  kann  es  aus  diesen  nicht  wieder 
zusammenstellen  und  beleben." 

Among  certain  microscopic  organisms  such  as  the  Rotifera  (which 
have  the  least  average  size  and  the  narrowest  range  of  size  of  all 
the  Metazoa),  we  are  still  more  palpably  struck  by  the  small  number 
of  cells  which  go  to  constitute  a  usually  complex  organ,  such  as 
kidney,  stomach  or  ovary ;  we  can  sometimes  number  them  in  a  few 

*  While  the  difference  in  cell-volume  is  vastly  less  than  that  between  the 
volumes,  and  very  much  less  also  than  that  between  the  surfaces,  of  the  respective 
animals,  yet  there  is  a  certain  difference ;  and  this  it  has  been  attempted  to  correlate 
with  the  need  for  each  cell  in  the  many-celled  ganglion  of  the  larger  animal  to 
possess  a  more  complex  "exchange-system"  of  branches,  for  intercommunication 
with  its  more  numerous  neighbours.  Another  explanation  is  based  on  the  fact 
that,  while  such  cells  as  continue  to  divide  throughout  life  tend  to  uniformity  of 
size  in  all  mammals,  those  which  do  not  do  so,  and  in  particular  the  ganglion  cells, 
continue  to  grow,  and  their  size  becomes,  therefore,  a  function  of  the  duration  of 
life.  Cf.  G.  Levi,  Studii  sulla  grandezza  delle  cellule.  Arch.  Jtal.  di  Anat.  e  di 
Embrioloq.  v,  p.  291,  1906;  cf.  also  A.  Berezowski,  Studien  liber  die  Zellgrosse, 
Arch.  f.  Zellforsch.  v,  pp.  375-384,  1910. 


II]  OF  THE  LEAST  OF  ORGANISMS  63 

units,  in  place  of  the  many  thousands  which  make  up  such  an  organ 
in  larger,  if  not  alwayc  higher,  animals.  We  have  already  spoken 
of  the  Fairy-flies,  a  few  score  of  which  would  hardly  weigh  down 
one  of  the  larger  rotifers,  and  a  hundred  thousand  would  weigh  less 
than  one  honey-bee.  Their  form  is  complex  and  their  httle  bodies 
exquisitely  beautiful;  but  I  feel  sure  that  their  cells  are  few,  and 
their  organs  of  great  histological  simplicity.  These  considerations 
help,  I  think,  to  shew  that,  however  important  and  advantageous 
the  subdivision  of  the  tissues  into  cells  may  be  from  the  construc- 
tional, or  from  the  dynamic,  point  of  view,  the  phenomenon  has 
less  fundamental  importance  than  was  once,  and  is  often  still, 
assigned  to  it. 

Just  as  Sachs  shewed  there  was  a  hmit  to  the  amount  of  cytoplasm 
which  could  gather  round  a  nucleus,  so  Boveri  has  demonstrated 
that  the  nucleus  itself  has  its  own  hmitations  of  size,  and  that,  in 
cell-division  after  fertilisation,  each  new  nucleus  has  the  same  size 
as  its  parent  nucleus*;  we  may  nowadays  transfer  the  statement 
to  the  chromosomes.  It  may  be  that  a  bacterium  lacks  a  nucleus 
for  the  simple  reason  that  it  is  too  small  to  hold  one,  and  that  the 
same  is  true  of  such  small  plants  as  the  Cyanophyceae,  or  blue-green 
algae.  Even  a  chromatophore  with  its  "pyrenoids"  seems  to  be 
impossible  below  a  certain  sizej. 

Always  then,  there  are  reasons,  partly  physiological  but  in  large 
part  purely  physical,  which  define  or  regulate  the  magnitude  of  the 
organism  or  the  cell.  And  as  we  have  already  found  definite 
Hmitations  to  the  increase  in  magnitude  of  an  organism,  let  us  now 
enquire  whether  there  be  not  also  a  lower  limit  below  which  the 
very  existence  of  an  organism  becomes  impossible. 

*  Boveri,  Zellen.studien,  V:  Ueber  die  Abhangigkeit  der  Kerngrosse  und 
Zellenzahl  von  der  Chromosomenzahl  der  Ausgangszellen.  Jena,  1905.  Cf.  also 
{int.  al.)  H.  Voss,  Kerngrossenverhaltnisse  in  der  Leber  etc.,  Ztschr.f.  Zellforschung, 
VII,  pp.  187-200,  1928. 

t  The  size  of  the  nucleus  may  be  affected,  even  determined,  by  the  number  of 
chromosomes  it  contains.  There  are  giant  race*  of  Oenothera,  Primula  and  Solanum 
whose  cell-nuclei  contain  twice  the  normal  number  of  chromosomes,  and  a  dwarf 
race  of  a  little  freshwater  crustacean,  Cyclops,  has  half  the  usual  number.  The 
cytoplasm  in  turn  varies  with  the  amount  of  nuclear  matter,  the  whole  cell  is 
unusually  large  or  unusually  small;  and  in  these  exceptional  cases  we  see  a  direct 
relation  between  the  size  of  the  organism  and  the  size  of  the  cell.  Cf.  {int.  al.) 
R.  P.  Gregory,  Proc.  Camb.  Phil  Soc.  xv,  pp.  239-246,  1909;  F.  Keeble,  Journ. 
of  Genetics,  ii,  pp.  163-188,  1912.  • 


64  ON  MAGNITUDE  [ch. 

A  bacillus  of  ordinary  size  is,  say,  1  /x  in  length.  The  length  (or 
height)  of  a  man  is  about  a  million  and  three-quarter  times  as  great, 
i.e.  1-75  metres,  or  1-75  x  10^ /x;  and  the  mass  of  the  man  is  in  the 
neighbourhood  of  5  x  10^®  (five  million,  million,  milHon)  times 
greater  than  that  of  the  bacillus.  If  we  ask  whether  there  may  not 
exist  organisms  as  much  less  than  the  bacillus  as  the  bacillus  is  less 
than  the  man,  it  is  easy  to  reply  that  this  is  quite  impossible,  for  we 
are  rapidly  approaching  a  point  where  the  question  of  molecular 
dimensions,  and  of  the  ultimate  divisibility  of  matter,  obtrudes 
itself  as  a  crucial  factor  in  the  case.  Clerk  Maxwell  dealt  with  this 
matter  seventy  years  ago,  in  his  celebrated  article  Atom"^.  KoUi 
(or  Colley),  a  Russian  chemist,  declared  in  1893  that  the  head  of  a 
spermatozoon  could  hold  no  more  than  a  few  protein  molecules ;  and 
Errera,  ten  years  later,  discussed  the  same  topic  with  great  ingenuity  f. 
But  it  needs  no  elaborate  calculation  to  convince  us  that  the  smaller 
bacteria  or  micrococci  nearly  approach  the  smallest  magnitudes 
which  we  can  conceive  to  have  an  organised  structure.  A  few  small 
bacteria  are  the  smallest  of  visible  organisms,  and  a  minute  species 
associated  with  influenza,  B.  pneumosinter,  is  said  to  be  the  least 
of  them  all.  Its  size  is  of  the  order  of  0-1 /x,  or  rather  less;  and 
here  we  are  in  close  touch  with  the  utmost  limits  of  microscopic 
vision,  for  the  wave-lengths  of  visible  light  run  only  from  about 
400  to  700 m/x.  The  largest  of  the  bacteria,  B.  megatherium,  larger 
than  the  well-known  B.  anthracis  of  splenic  fever,  has  much  the 
same  proportion  to  the  least  as  an  elephant  to  a  guinea-pig  {. 

Size  of  body  is  no  mere  accident.  Man,  respiring  as  he  does, 
cannot  be  as  small  as  an  insect,  nor  vice  versa;  only  now  and  then, 
as  in  the  Goliath  beetle,  do  the  sizes  of  mouse  and  beetle  meet  and 
overlap.  The  descending  scale  of  mammals  stops  short  at  a  weight 
of  about  5  grams,  that  of  beetles  at  a  length  of  about  half  a  milli- 
metre, and  every  group  of  animals  has  its  upper  and  its  lower 
limitations  of  size.  So,  not  far  from  the  lower  limit  of  our  vision, 
does  the  long  series  of  bacteria  come  to  an  end.  There  remain  still 
smaller  particles  which  the  ultra-microscope  in  part  reveals;    and 

*  Encyclopaedia  Britannica,  9th  edition,  1875. 

f  Leo  Errera,  Siir  la  limite  de  la  petitesse  des  organismes,  BvlL  Soc.  Roy.  des 
Sc.  me'd.  et  nat.  de  Bruxelles,  1903;   Recueil  d'osnvres  {Physiologie  gen^rale),  p.  325. 

I  Cf.  A.  E.  Boycott,  The  transition  from  live  to  dead,  Proc.  R.  Soc.  of  Medicine, 
XXII  {Pathology),  pp.  55-69,  1928. 


II]  OF  MOLECULAR  MAGNITUDES  65 

here  or  hereabouts  are  said  to  come  the  so-called  viruses  or  "filter- 
passers,"  brought  within  our  ken  by  the  maladies,  such  as  hydro- 
phobia, or  foot-and-mouth  disease,  or  the  mosaic  diseases  of  tobacco 
and  potato,  to  which  they  give  rise.  These  minute  particles,  of  the 
order  of  one-tenth  the  diameter  of  our  smallest  bacteria,  have  no 
diffusible  contents,  no  included  water — whereby  they  differ  from 
every  living  thing.  They  appear  to  be  inert  colloidal  (or  even 
crystalloid)  aggregates  of  a  nucleo-protein,  of  perhaps  ten  times  the 
diameter  of  an  ordinary  protein-molecule,  and  not  much  larger  than 
the  giant  molecules  of  haemoglobin  or  haemocyanin  *. 

Bejerinck  called  such  a  virus  a  contagium  vivum;  "infective 
nucleo-protein"  is  a  newer  name.  We  have  stepped  down,  by  a 
single  step,  from  Hving  to  non-Hving  things,  from  bacterial  dimen- 
sions to  the  molecular  magnitudes  of  protein  chemistry.  And  we 
begin  to  suspect  that  the  virus-diseases  are  not  due  to  an  "organism, 
capable  of  physiological  reproduction  and  multiphcation,  but  to  a 
mere  specific  chemical  substance,  capable  of  catalysing  pre-existing 
materials  and  thereby  producing  more  and  more  molecules  hke 
itself.  The  spread  of  the  virus  in  a  plant  would  then  be  a  mere 
autocatalysis,  not  involving  the  transport  of  matter,  but  only  a 
progressive  change  of  state  in  substances  already  there  f." 

But,  after  all,  a  simple  tabulation  is  all  we  need  to  shew  how 
nearly  the  least  of  organisms  approach  to  molecular  magnitudes. 
The  same  table  will  suffice  to  shew  how  each  main  group  of  animals 
has  its  mean  and  characteristic  size,  and  a  range  on  either  side, 
sometimes  greater  and  sometimes  less. 

Our  table  of  magnitudes  is  no  mere  catalogue  of  isolated  facts, 
but  goes  deep  into  the  relation  between  the  creature  and  its  world. 
A  certain  range,  and  a-  narrow  one,  contains  mouse  and  elephant, 
and  all  whose  business  it  is  to  walk  and  run ;   this  is  our  own  world, 

*  Cf.  Svedberg,  Journ.  Am.  Chem.  Soc.  XLviir,  p.  30,  1926.  According  to  the 
Foot-and-Mouth  Disease  Research  Committee  {oth  Report,  1937),  the  foot-and- 
mouth  virus  has  a  diameter,  determined  by  graded  filters,  of  8-12m/i;  while 
Kenneth  Smith  and  W.  D.  MacClement  {Proc.  R.S.  (B),  cxxv,  p.  296,  1938)  calculate 
for  certain  others  a  diameter  of  no  more  than  4m^,  or  less  than  a  molecule  of 
haemocyanin. 

t  H.  H.  Dixon,  Croonian  lecture  on  the  transport  of  substances  in  plants, 
Proc.  R.S.  (B),  vol.  cxxv,  pp.  22,  23,  1938. 


66 


ON  MAGNITUDE 


[CH. 


with  whose  dimensions  our  hves,  our  hmbs,  our  senses  are  in  tune. 
The  great  whales  grow  out  of  this  range  by  throwing  the  burden 
of  their  bulk  upon  the  waiters;  the  dinosaurs  wallowed  in  the  swamp, 
and  the  hippopotamus,  the  sea-elephant  and  Steller's  great  sea-cow 
pass  or  passed  their  hves  in  the  rivers  or  the  sea.     The  things  which 


Linear  dimensions  of  organisms,  and  other  objects 


cm. 

(10,000  km.) 

10' 

A  quadrant  of  the  earth's  circumference 

(1000  km.) 

10« 
105 
10* 

Orkney  to  Land's  End 
Mount  Everest 

(km.) 

103 

102 

Giant  trees:   Sequoia 
Large  whale 
Basking  shark 

101 

Elephant;   ostrich;   man 

(metre) 

10" 
10-1 

Dog;  rat;  eagle 

Small  birds  and  mammals;   large  insects 

(cm.) 

10-2 

Small  insects;   minute  fish 

(mm.) 

10-3 

Minute  insects 

10-* 

Protozoa;   pollen -grains 

-  Cells 

10-5 

Large  bacteria;   human  blood-corpuscles 

(micron,  /z,) 

io-« 

10-' 

Minute  bacteria 

Limit  of  microscopic  vision 

io-« 

Viruses,  or  filter-passers                 r.  n    j        *•  i 
Giant  albuminoids,  casein,  etc.f    ^^^^^^  P^^^^^^^^ 
Starch-molecule 

(m/ii) 

io-» 

Water-molecule 

(Angstrom  unit) 

10-10 

fly  are  smaller  than  the  things  which  walk  and  run ;  the  flying  birds 
are  never  as  large  as  the  larger  mammals,  the  lesser  birds  and 
mammals  are  much  of  a  muchness,  but  insects  come  down  a  step 
in  the  scale  and  more.  The  lessening  influence  of  gravity  facilitates 
flight,  but  makes  it  less  easy  to  walk  and  run;  first  claws,  then 
hooks  and  suckers  and  glandular  hairs  help  to  secure  a  foothold, 


II]  OF  SCALES  OF  MAGNITUDE  67 

until  to  creep  upon  wall  or  ceiling  becomes  as  easy  as  to  walk  upon 
the  ground.  Fishes,  by  evading  gravity,  increase  their  range  of 
magnitude  both  above  and  below  that  of  terrestrial  animals.  Smaller 
than  all  these,  passing  out  of  our  range  of  vision  and  going  down  to 
the  least  dimensions  of  hving  things,  are  protozoa,  rotifers,  spores, 
pollen-grains*  and  bacteria.  All  save  the  largest  of  these  float 
rather  than  swim;  they  are  buoyed  up  by  air  or  water,  and  fall 
(as  Stokes's  law  explains)  with  exceeding  slowness. 

There  is  a  certain  narrow  range  of  magnitudes  where  (as  we  have 
partly  said)  gravity  and  surface  tension  become  comparable  forces, 
nicely  balanced  with  one  another.  Here  a  population  of  small 
plants  and  animals  not  only  dwell  in  the  surface  waters  but  are 
bound  to  the  surface  film  itself — the  whirligig  beetles  and  pond- 
skaters,  the  larvae  of  gnat  and  mosquito,  the  duckweeds  (Lemna), 
the  tiny  Wolffia,  and  Azolla;  even  in  mid-ocean,  one  small  insect 
(Halobates)  retains  this  singular  habitat.  It  would  be  a  long  story 
to  tell  the  various  ways  in  which  surface-tension  is  thus  taken  full 
advantage  of.  Gravitation  not  only  Hmits  the  magnitude  but 
controls  the  form  of  things.  With  the  help  of  gravity  the  quadruped 
has  its  back  and  its  belly,  and  its  limbs  upon  the  ground ;  its  freedom 
of  motion  in  a  plane  perpendicular  to  gravitational  force ;  its  sense 
of  fore-and-aft,  its  head  and  tail,  its  bilateral  symmetry.  Gravitation 
influences  both  our  bodies  and  our  minds.  We  owe  to  it  our  sense 
of  the  vertical,  our  knowledge  of  up-and-down;  our  conception  of 
the  horizontal  plane  on  which  we  stand,  and  our  discovery  of  two 
axes  therein,  related  to  the  vertical  as  to  one  another;  it  was  gravity 
which  taught  us  to  think  of  three-dimensional  space.  Our  archi- 
tecture is  controlled  by  gravity,  but  gravity  has  less  influence  over 
the  architecture  of  the  bee ;  a  bee  might  be  excused,  might  even  be 
commended,  if  it  referred  space  to  four  dimensions  instead  of  three !  t 
The  plant  has  its  root  and  its  stem",    but  about  this  vertical  or 

*  Pollen-grains,  like  protozoa,  have  a  considerable  range  of  magnitude.  The 
largest,  such  as  those  of  the  pumpkin,  are  about  200/x  in  diameter;  these  have  to 
be  carried  by  insects,  for  they  are  above  the  level  of  Stokes's  law,  and  no  longer 
float  upon  the  air.  The  smallest  pollen-grains,  such  as  those  of  the  forget-me-not, 
are  about  4^  /x  in  diameter  (Wodehouse). 

f  Corresponding,  that  is  to  say,  to  the  four  axes  which,  meeting  in  a  pfoint,  make 
co-equal  angles  (the  so-called  tetrahedral  angles)  one  with  another,  as  do  the  basal 
angles  of  the  honeycomb.     (See  below,  chap,  vu.) 


68  ON  MAGNITUDE  [ch. 

gravitational  axis  its  radiate  symmetry  remains,  undisturbed  by 
directional  polarity,  save  for  the  sun.  Among  animals,  radiate 
symmetry  is  confined  to  creatures  of  no  great  size ;  and  some  form 
or  degree  of  spherical  symmetry  becomes  the  rule  in  the  small  world 
of  the  protozoon — unless  gravity  resume  its  sway  through  the  added 
burden  of  a  shell.  The  creatures  which  swim,  walk  or  run,  fly, 
creep  or  float  are,  so  to  speak,  inhabitants  and  natural  proprietors 
of  as  many  distinct  and  all  but  separate  worlds.  Humming-bird 
and  hawkmoth  may,  once  in  a  way,  be  co-tenants  of  the  same 
world;  but  for  the  most  part  the  mammal,  the  bird,  the  fish,  the 
insect  and  the  small  life  of  the  sea,  not  only  have  their  zoological 
distinctions,  but  each  has  a  physical  universe  of  its  own.  The 
world  of  bacteria  is  yet  another  world  again,  and  so  is  the  world  of 
colloids ;  but  through  these  small  Lilliputs  we  pass  outside  the  range 
of  hving  things. 

What  we  call  mechanical  principles  apply  to  the  magnitudes 
among  which  we  are  at  home;  but  lesser  worlds  are  governed  by 
other  and  appropriate  physical  laws,  of  capillarity,  adsorption  and 
electric  charge.  There  are  other  worlds  at  the  far  other  end  of  the 
scale,  in  the  uttermost  depths  of  space,  whose  vast  magnitudes  lie 
within  a  narrow  range.  When  the  globular  star-clusters  are  plotted 
on  a  curve,  apparent  diameter  against  estimated  distance,  the 
curve  is  a  fair  approximation  to  a  rectangular  hyperbola;  which 
means  that,  to  the  same  rough  approximation,  the  actual  diameter 
is  identical  in  them  all*. 

It  is  a  remarkable  thing,  worth  pausing  to  reflect  on,  that  we  can 
pass  so  easily  and  in  a  dozen  fines  from  molecular  magnitudes  f  to 
the  dimensions  of  a  Sequoia  or  a  whale.  Addition  and  subtraction, 
the  old  arithmetic  of  the  Egyptians,  are  not  powerful  enough  for 
such  an  operation;  but  the  story  of  the  grains  of  wheat  upon  the 
chessboard  shewed  the  way,  and  Archimedes  and  Napier  elaborated 

*  See  Harlow  Shapley  and  A.  B.  Sayer,  The  angular  diameters  of  globular 
clusters,  Proc.  Nat.  Acad,  of  Sci.  xxi,  pp.  593-597,  1935.  The  same  is  approxi- 
mately true  of  the  spiral  nebulae  also. 

t  We  may  call  (after  Siedentopf  and  Zsigmondi)  the  smallest  visible  particles 
microns,  such  for  instance  as  small  bacteria,  or  the  fine  particles  of  gum-mastich 
in  suspension,  measuring  0-5  to  1-0/x;  sub-microns  are  those  revealed  by  the  ultra- 
microscope,  such  as  particles  of  colloid  gold  (2-15m/Lt),  or  starch-moleculea  (5m/x); 
amicrons,  under  Im^,  are  not  perceptible  by  either  method.  A  water-molecule 
measures,  probably,  about  0-1  m/i. 


II]  OF  THIN  FILMS  69 

the  arithmetic  of  multipHcation.  So  passing  up  and  down  by  easy- 
steps,  as  Archimedes  did  when  he  numbered  the  sands  of  the  sea, 
we  compare  the  magnitudes  of  the  great  beasts  and  the  small,  of 
che  atoms  of  which  they  are  made,  and  of  the  world  in  which  they 
dwell*. 

While  considerations  based  on  the  chemical  composition  of  the 
organism  have  taught  us  that  there  must  be  a  definite  lower  hmit 
to  its  magnitude,  other  considerations  of  a  purely  physical  kind  lead 
us  to  the  same  conclusion.  For  our  discussion  of  the  principle  of 
similitude  has  already  taught  us  that  long  before  we  reach  these 
all  but  infinitesimal  magnitudes  the  dwindling  organism  will  have 
experienced  great  changes  in  all  its  physical  relations,  and  must  at 
length  arrive  at  conditions  surely  incompatible  with  life,  or  what  we 
understand  as  life,  in  its  ordinary  development  and  manifestation. 

We  are  told,  for  instance,  that  the  powerful  force  of  surface-tension, 
or  capillarity,  begins  to  act  within  a  range  of  about  1/500,000  of  an 
inch,  or  say  0-05 />t.  A  soap  film,  or  a  film  of  oil  on  water,  may  be 
attenuated  to  far  less  magnitudes  than  this;  the  black  spots  on  a 
soap  bubble  are  known,  by  various  concordant  methods  of  measure- 
ment, to  be  only  about  6x  10~'  cm.,  or  about  6m/x  thick,  and  Lord 
Rayleigh  and  M.  Devaux  have  obtained  films  of  oil  of  2mjLt,  or  even 
1  m/x  in  thickness.  But  while  it  is  possible  for  a  fluid  film  to  exist 
of  these  molecular  dimensions,  it  is  certain  that  long  before  we 
reach  these  magnitudes  there  arise  conditions  of  which  we  have 
little  knowledge,  and  which  it  is  not  easy  to  imagine.  A  bacillus 
lives  in  a  world,  or  on  the  borders  of  a  world,  far  other  than  our 
own,  and  preconceptions  drawn  from  our  experience  are  not  valid 
there.  Even  among  inorganic,  non-living  bodies,  there  comes  a 
certain  grade  of  minuteness  at  which  the  ordinary  properties  become 
modified.  For  instance,  while  under  ordinary  circumstances  crystal- 
lisation starts  in  a  solution  about  a  minute  sohd  fragment  or  crystal 

*  Observe  that,  following  a  common  custom,  we  have  only  used  a  logarithmic 
scale  for  the  round  numbers  representing  powers  of  ten,  leaving  the  interspaces 
between  these  to  be  filled  up,  if  at  all,  by  ordinary  numbers.  There  is  nothing 
to  prevent  us  from  using  fractional  indices,  if  we  please,  throughout,  and  calling 
a  blood-corpuscle,  for  instance,  10~^"^  cm.  in  diameter,  a  man  lO^'^^  cm.  high,  or 
Sibbald's  Rorqual  lOi"*^  metres  long.  This  method,  implicit  in  that  of  Napier  of 
Merchiston,  was  first  set  forth  by  Wallis,  in  his  Arithmetica  infinitorum. 


70  ON  MAGNITUDE  [ch. 

of  the  salt,  Ostwald  has  shewn  that  we  may  have  particles  so  minute 
that  they  fail  to  serve  as  a  nucleus  for  crystalUsation — which  is  as 
much  as  to  say  that  they  are  too  small  to  have  the  form  and  pro- 
perties of  a  "crystal."  And  again,  in  his  thin  oil-films,  Lord 
Rayleigh  noted  the  striking  change  of  physical  properties  which 
ensues  when  the  film  becomes  attenuated  to  one,  or  something  less 
than  one,  close-packed  layer  of  molecules,  and  when,  in  short,  it  no 
longer  has  the  properties  of  matter  in  mass. 

These  attenuated  films  are  now  known  to  be  "monomolecular,"  the 
long-chain  molecules  of  the  fatty  acids  standing  close-packed,  like  the  cells 
of  a  honeycomb,  and  the  film  being  just  as  thick  as  the  molecules  are  long. 
A  recent  determination  makes  the  several  molecules  of  oleic,  palmitic  and 
stearic  acids  measure  10-4,  14-1  and  15- 1  cm.  in  length,  and  in  breadth  7-4, 
6-0  and  5-5  cm.,  all  by  10~^:  in  good  agreement  with  Lord  Rayleigh  and 
Devaux's  lowest  estimates  (F.  J.  Hill,  Phil.  Mag.  1929,  pp.  940-946).  But 
it  has  since  been  shewn  that  in  aliphatic  substances  the  long-chain  molecules 
are  not  erect,  but  inclined  to  the  plane  of  the  film ;  that  the  zig-zag  constitution 
of  the  molecules  permits  them  to  interlock,  so  giving  the  film  increased 
stability ;  and  that  the  interlock  may  be  by  means  of  a  first  or  second  zig-zag, 
the  measured  area  of  the  film  corresponding  precisely  to  these  two  dimorphic 
arrangements.  (Cf.  C.  G.  Lyons  and  E.  K.  Rideal,  Proc.  R.S.  (A),  cxxvin, 
pp.  468-473,  1930.)  The  film  may  be  lifted  on  to  a  polished  surface  of  metal, 
or  even  on  a  sheet  of  paper,  and  one  monomolecular  layer  so  added  to  another; 
even  the  complex  protein  molecule  can  be  unfolded  to  form  a  film  one  amino- 
acid  molecule  thick.  The  whole  subject  of  monomolecular  layers,  the  nature 
of  the  film,  whether  condensed,  expanded  or  gaseous,  its  astonishing  sensitive- 
ness to  the  least  impurities,  apd  the  manner  of  spreading  of  the  one  liquid 
over  the  other,  has  become  of  great  interest  and  importance  through  the  work 
of  Irving  Langmuir,  Devaux,  N.  K.  Adam  and  others,  and  throws  new  light 
on  the  whole  subject  of  molecular  magnitudes*. 

The  surface-tension  of  a  drop  (as  Laplace"  conceived  it)  is  the 
cumulative  effect,  the  statistical  average,  of  countless  molecular 
attractions,  but  we  are  now  entering  on  dimensions  where  the 
molecules  are  fewf.  The  free  surface-energy  of  a  body  begins  to 
vary  with  the  radius,  when  that  radius  is  of  an  order  comparable 
to  inter-molecular  distances;  and  the  whole  expression  for  such 
energy  tends  to  vanish  away  when  the  radius  of  the  drop  or  particle 
is  less  than  O-Olfx,  or  lOm^it.     The  quahties  and  properties  of  our 

*  Cf.  (int.  al.)  Adam,  Physics  and  Chemistry  of  Surfaces,  1930;  Irving  Langmuir, 
Proc.  R.S.  (A),  CLXX,  1939. 

t  See  a  very  interesting  paper  by  Fred  Vies,  Introduction  a  la  physique  bac- 
terienne,  Revue  Sclent.  11  juin  1921.  Cf.  also  N.  Rashevsky,  Zur  Theorie  d. 
spontanen  Teilung  von  mikroskopischen  Tropfen,  Ztschr.f.  Physik,  xlvi,  p.  578, 1928. 


II]  OF  MINUTE   MAGNITUDES  71 

particle  suffer  an  abrupt  change  here;  what  then  can  we  attribute, 
in  the  way  of  properties,  to  a  corpuscle  or  organism  as  small  or 
smaller  than,  say,  0-05  or  0-03 /x?  It  must,  in  all  probability,  be  a 
homogeneous  structureless  body,  composed  of  a  very  small  number 
of  albumenoid  or  other  molecules.  Its  vital  properties  and  functions 
must  be  extremely  limited;  its  specific  outward  characters,  even  if 
we  could  see  it,  must  be  nil;  its  osmotic  pressure  and  exchanges 
must  be  anomalous,  and  under  molecular  bombardment  they  may 
be  rudely  disturbed;  its  properties  can  be  Httle  more  than  those  of 
an  ion-laden  corpuscle,  enabling  it  to  perform  this  or  that  specific 
chemical  reaction,  to  effect  this  or  that  disturbing  influence,  or 
produce  this  or  that  pathogenic  effect.  Had  it  sensation,  its  ex- 
periences would  be  strange  indeed;  for  if  it  could  feel,  it  would  regard 
a  fall  in  temperature  as  a  movement  of  the  molecules  around,  and 
if  it  could  see  it  would  be  surrounded  with  light  of  many  shifting 
colours,  like  a  room  filled  with  rainbows. 

The  dimensions  of  a  cilium  are  of  such  an  order  that  its  substance 
is  mostly,  if  not  all,  under  the  pecuHar  conditions  of  a  surface-layer, 
and  surface-energy  is  bound  to  play  a  leading  part  in  cihary  action. 
A  cilium  or  flagellum  is  (as  it  seems  to  me)  a  portion  of  matter  in 
a  state  sui  generis,  with  properties  of  its  own,  just  as  the  film  and  the 
jet  have  theirs.  And  just  as  Savart  and  Plateau  have  told  us  about 
jets  and  films,  so  will  the  physicist  some  day  explain  the  properties 
of  the  cilium  and  flagellum.  It  is  certain  that  we  shall  never 
understand  these  remarkable  structures  so  long  as  we  magnify 
them  to  another  scale,  and  forget  that  new  and  pecuhar  physical 
properties  are  associated  with  the  scale  to  which  they  belong*. 

As  Clerk  Maxwell  put  it,  "molecular  science  sets  us  face  to  face 
with  physiological  theories.  It  forbids  the  physiologist  to  imagine 
that  structural  details  of  infinitely  small  dimensions  (such  as  Leibniz 
assumed,  one  within  another,  ad  infinitum)  can  furnish  an  explana- 
tion of  the  infinite  variety  which  exists  in  the  properties  and  functions 
of  the  most  minute  organisms."  And  for  this  reason  Maxwell 
reprobates,  with  not  undue  severity,  those  advocates  of  pangenesis 

*  The  cilia  on  the  gills  of  bivalve  molluscs  are  of  exceptional  size,  measuring 
from  say  20  to  120^  long.  They  are  thin  triangular  plates,  rather  than  filaments; 
they  are  from  4  to  lO/x  broad  at  the  base,  but  less  than  1/x  thick.  Cf.  D.  Atkins. 
Q.J. M.S.,  1938,  and  other  papers. 


72  ON  MAGNITUDE  [ch. 

and  similar  theories  of  heredity,  who  "would  place  a  whole  world 
of  wonders  within  a  body  so  small  and  so  devoid  of  visible  structure 
as  a  germ."  But  indeed  it  scarcely  needed  Maxwell's  criticism  to 
shew  forth  the  immense  physical  difficulties  of  Darwin's  theory  of 
pangenesis:  which,  after  all,  is  as  old  as  Democritus,  and  is  no  other 
than  that  Promethean  particula  undique  desecta  of  which  we  have 
read,  and  at  which  we  have  smiled,  in  our  Horace. 

There  are  many  other  ways  in  which,  when  we  make  a  long 
excursion  into  space,  we  find  our  ordinary  rules  of  physical  behaviour 
upset.  A  very  familiar  case,  analysed  by  Stokes,  is  that  the 
viscosity  of  the  surrounding  medium  has  a  relatively  powerful  effect 
upon  bodies  below  a  certain  size.  A  droplet  of  water,  a  thousandth 
of  an  inch  (25  ju,)  in  diameter,  cannot  fall  in  still  air  quicker  than 
about  an  inch  and  a  half  per  second;  as  its  size  decreases,  its 
resistance  varies  as  the  radius,  not  (as  with  larger  bodies)  as  the 
surface;  and  its  "critical"  or  terminal  velocity  varies  as  the 
square  of  the  radius,  or  as  the  surface  of  the  drop.  A  minute 
drop  in  a  misty  cloud  may  be  one-tenth  that  size,  and  will  fall  a 
hundred  times  slower,  say  an  inch  a  minute;  and  one  again  a  tenth 
of  this  diameter  (say  0-25 />t,  or  about  twice  as  big  as  a  small  micro- 
coccus) will  scarcely  fall  an  inch  in  two  hours*.  Not  only  do 
dust-particles,  spores  f  and  bacteria  fall,  by  reason  of  this  principle, 
very  slowly  through  the  air,  but  all  minute  bodies  meet  with  great 
proportionate  resistance  to  their  movements  through  a  fluid.  In 
salt  water  they  have  the  added  influence  of  a  larger  coefficient  of 
friction  than  in  fresh  J ;  and  even  such  comparatively  large  organisms 
as  the  diatoms  and  the  foraminifera,  laden  though  they  are  with  a 
heavy  shell  of  flint  or  lime,  seem  to  be  poised  in  the  waters  of  the 
ocean,  and  fall  with  exceeding  slowness. 

*  The  resistance  depends  on  the  radius  of  the  particle,  the  viscosity,  and  the 
rate  of  fall  ( V) ;  the  eifective  weight  by  which  this  resistance  is  to  be  overcome 
depends  on  gravity,  on  the  density  of  the  particle  compared  with  that  of  the 
medium,  and  on  the  mass,  which  varies  as  r^.  Resistance  =A;rF,  and  effective 
weight  =  )fcV;  when  these  two  equal  one  another  we  have  the  critical  or  terminal 
velocity,  and  Vccr^. 

t  A.  H.  R.  BuUer  found  the  spores  of  a  fungus  (CoUybia),  measuring  5x3/i, 
to  fall  at  the  rate  of  half  a  millimetre  per  second,  or  rather  more  than  an  inch 
a  minute;   Studies  on  Fungi,  1909. 

X  Cf.  W.  Krause,  Biol.  Centralbl.  i,  p.  578,  1881;  Fliigel,  Meteorol  Ztschr.  1881, 
p.  321. 


II]  OF  STOKES'S  LAW  73 

When  we  talk  of  one  thing  touching  another,  there  may  yet  be 
a  distance  between,  not  only  measurable  but  even  large  compared 
with  the  magnitudes  we  have  been  considering.  Two  polished 
plates  of  glass  or  steel  resting  on  one  another  are  still  about  4/x 
apart — the  average  size  of  the  smallest  dust;  and  when  all  dust- 
particles  are  sedulously  excluded,  the  one  plate  sinks  slowly  down 
to  within  O-S/jl  of  the  other,  an  apparent  separation  to  be  accounted 
for  by  minute  irregularities  of  the  polished  surfaces*. 

The  Brownian  movement  has  also  to  be  reckoned  with — that 
remarkable  phenomenon  studied  more  than  a  century  ago  by  Robert 
Brown  f,  Humboldt's /ac?7e  princeps  botanicorum,  and  discoverer  of 
the  nucleus  of  the  cell  J.  It  is  the  chief  of  those  fundamental 
phenomena  which  the  biologists  have  contributed,  or  helped  to 
contribute,  to  the  science  of  physics. 

The  quivering  motion,  accompanied  by  rotation  and  even  by 
translation,  manifested  by  the  fine  granular  particle  issuing  from  a 
crushed  pollen-grain,  and  which  Brown  proved  to  have  no  vital 
significance  but  to  be  manifested  by  all  minute  particles  whatsoever, 
was  for  many  years  unexplained.  Thirty  years  and  more  after  Brown 
wrote,  it  was  said  to  be  "due,  either  directly  to  some  calorical 
changes  continually  taking  place  in  the  fluid,  or  to  some  obscure 
chemical  action  between  the  solid  particles  and  the  fluid  which  is 
indirectly   promoted   by   heat§."     Soon    after   these    words    were 


*  Cf.  Hardy  and  Nottage,  Proc.  R.S.  (A),  cxxviii,  p.  209,  1928;  Baston  and 
Bowden,  ibid,  cxxxiv,  p.  404,  1931. 

t  A  Brief  Description  of  Microscopical  Observations.  .  .on  the  Particles  contained 
in  the  Pollen  of  Plants;  and  on  the  General  Existence  of  Active  Molecules  in  Organic 
and  Inorganic  Bodies,  London,  1828.  See  also  Edinb.  Netv  Philosoph.  Journ.  v, 
p.  358,  1828;  Edinb.  Journ.  of  Science,  i,  p.  314,  1829;  Ann.  Sc.  Nat.  xiv,  pp.  341- 
362,  1828;  etc.  The  Brownian  movement  was  hailed  by  some  as  supporting 
Leibniz's  theory  of  Monads,  a  theory  once  so  deeply  rooted  and  so  widely  believed 
that  even  under  Schwann's  cell-theory  Johannes  Miiller  and  Henle  spoke  of 
the  cells  as  "organische  Monaden";  cf.  Emit  du  Bois  Reymond,  Leibnizische 
Gedanken  in  der  neueren  Naturwissenschaft,  Monatsber.  d.  k.  Akad.  Wiss.,  Berlin, 
1870. 

J  The  "nucleus"  was  first  seen  in  the  epidermis  of  Orchids;  but  "this  areola, 
or  nucleus  of  the  cell  as  perhaps  it  might  be  termed,  is  not  confined  to  the 
epidermis,"  etc.  See  his  paper  on  Fecundation  in  Orchideae  and  Asclepiadae, 
Trans.  Linn.  Soc.  xvi,  1829-33,  also  Proc.  Linn.  Soc.  March  30,  1832. 

§  Carpenter,  The  Microscope,  edit.  1862,  p.  185. 


74  ON  MAGNITUDE  [ch. 

written  it  was  ascribed  by  Christian  Wiener  *  to  molecular  move- 
ments within  the  fluid,  and  was  hailed  as  visible  proof  of  the 
atomistic  (or  molecular)  constitution  of  the  same.  We  now  know 
that  it  is  indeed  due  to  the  impact  or  bombardment  of  molecules 
upon  a  body  so  small  that  these  impacts  do  not  average  out,  for 
the  moment,  to  approximate  equality  on  all  sides  f.  The  movement 
becomes  manifest  with  particles  of  somewhere  about  20 /x,  and  is 
better  displayed  by  those  of  about  10 /x,  and  especially  well  by 
certain  colloid  suspensions  or  emulsions  whose  particles  are  just 
below  1/Lt  in  diameter  {.  The  bombardment  causes  our  particles  to 
behave  just  hke  molecules  of  unusual  size,  and  this  behaviour  is 
manifested  in  several  ways§.  Firstly,  we  have  the  quivering 
movement  of  the  particles;  secondly,  their  movement  backwards 
and  forwards,  in  short,  straight  disjointed  paths;  thirdly,  the 
particles  rotate,  and  do  so  the  more  rapidly  the  smaller  they  are: 
and  by  theory,  confirmed  by  observation,  it  is  found  that  particles 
of  IjjL  in  diameter  rotate  on  an  average  through  100°  a  second, 
while  particles  of  13/x  turn  through  only  14°  a  minute.  Lastly,  the 
very  curious  result  appears,  that  in  a  layer  of  fluid  the  particles  are 
not  evenly  distributed,  nor  do  they  ever  fall  under  the  influence  of 
gravity  to  the  bottom.  For  here  gravity  and  the  Brownian  move- 
ment are  rival  powers,  striving  for  equilibrium;  just  as  gravity  is 
opposed  in  the  atmosphere  by  the  proper  motion  of  the  gaseous 
molecules.  And  just  as  equihbrium  is  attained  in  the  atmosphere 
when  the  molecules  are  so  distributed  that  the  density  (and  therefore 
the  number  of  molecules  per  unit  volume)  falls  oif  in  geometrical 

*  In  Poggendorffs  Annalen,  cxviii,  pp.  79-94,  1863.  For  an  account  of  this 
remarkable  man,  see  Naturmssensehaften,  xv,  1927;  cf.  also  Sigraund  Exner, 
Ueber  Brown's  Molecularbewegung,  Sitzungsher.  kk.  Akad.  Wien,  lvi,  p.  116,  1867. 

t  Perrin,  Les  preuves  de  la  realite  moleculaire,  Ann.  de  Physique,  xvii,  p.  549, 
1905;  XIX,  p.  571,  1906.  The  actual  molecular  collisions  are  unimaginably 
frequent;   we  see  only  the  residual  fluctuations. 

J  Wiener  was  struck  by  the  fact  that  the  phenomenon  becomes  conspicuous 
just  when  the  size  of  the  particles  becomes  comparable  to  that  of  a  wave-length 
of  light. 

§  For  a  full,  but  still  elementary,  account,  see  J.  Perrin,  Les  Atomes;  cf.  also 
Th.  Svedberg,  Die  Existenz  der  Molekiile,  1912;  R.  A.  Millikan,  The  Electron, 
1917,  etc.  The  modern  literature  of  the  Brownian  movement  (by  Einstein,  Perrin, 
de  Broglie,  Smoluchowski  and  Millikan)  is  very  large,  chiefly  owing  to  the  value 
which  the  phenomenon  is  shewn  to  have  in  determining  the  size  of  the  atom  or 
the  charge  on  an  electron,  and  of  giving,  as  Ostwald  said,  experimental  proof  of 
the  atomic  theory. 


II]  OF  THE  BROWNIAN  MOVEMENT  75 

progression  as  we  ascend  to  higher  and  higher  layers,  so  is  it  with 
our  particles  within  the  narrow  Umits  of  the  little  portion  of  fluid 
under  our  microscope. 

It  is  only  in  regard  to  particles  of  the  simplest  form  that  these 
phenomena  have  been  theoretically  investigated*,  and  we  may  take 
it  as  certain  that  more  complex  particles,  such  as  the  twisted  body 
of  a  Spirillum,  would  shew  other  and  still  more  comphcated  mani- 
festations. It  is  at  least  clear  that,  just  as  the  early  microscopists 
in  the  days  before  Robert  Brown  never  doubted  but  that  these 
phenomena  were  purely  vital,  so  we  also  may  still  be  apt  to  confuse, 
in  certain  cases,  the  one  phenomenon  with  the  other.  We  cannot, 
indeed,  without  the  most  careful  scrutiny,  decide  whether  the 
movements  of  our  minutest  organisms  are  intrinsically  "vital"  (in 
the  sense  of  being  beyond  a  physical  mechanism,  or  working  model) 
or  not.  For  example,  Schaudinn  has  suggested  that  the  undulating 
movements  of  Spirochaete  pallida  must  be  due  to  the  presence  of  a 
minute,  unseen,  "undulating  membrane";  and  Doflein  says  of  the 
same  species  that  "sie  verharrt  oft  mit  eigenthiimlich  zitternden 
Bewegungen  zu  einem  Orte."  Both  movements,  the  trembling  or 
quivering  movement  described  by  Doflein,  and  the  undulating  or 
rotating  movement  described  by  Schaudinn,  are  just  such  as  may 
be  easily  and  naturally  interpreted  as  part  and  parcel  of  the  Brownian 
phenomenon. 

While  the  Brownian  movement  may  thus  simulate  in  a  deceptive 
way  the  active  movements  of  an  organism,  the  reverse  statement 
also  to  a  certain  extent  holds  good.  One  sometimes  Hes  awake  of 
a  summer's  morning  watching  the  flies  as  they  dance  under  the 
ceiling.  It  is  a  very  remarkable  dance.  The  dancers  do  not  whirl  or 
gyrate,  either  in  company  or  alone;  but  they  advance  and  retire; 
they  seem  to  jostle  and  rebound;  between  the  rebounds  they  dart 
hither  or  thither  in  short  straight  snatches  of  hurried  flight,  and 
turn  again  sharply  in  a  new  rebound  at  the  end  of  each  little  rushf. 

*  Cf.  R.  Gans,  Wie  fallen  Stabe  und  Scheiben  in  einer  reibenden  Fliissigkeit? 
Miinchener  Bericht,  1911,  p.  191;  K.  Przibram,  Ueber  die  Brown'sche  Bewegung 
nicht  kugelformiger  Teilchen,  Wiener  Bericht,  1912,  p.  2339;   1913,  pp.  1895-1912. 

t  As  Clerk  Maxwell  put  it  to  the  British  Association  at  Bradford  in  1873,  "We 
cannot  do  better  than  observe  a  swarm  of  bees,  where  every  individual  bee  is 
flying  furiously,  first  in  one  direction  and  then  in  another,  while  the  swarm  as 
a  whole  is  either  at  rest  or  sails  slowly  through  the  air." 


76  ON  MAGNITUDE  [ch. 

Their  motions  are  erratic,  independent  of  one  another,  and 
devoid  of  common  purpose*.  This  is  nothing  else  than  a 
vastly  magnified  picture,  or  simulacrum,  of  the  Brownian  move- 
ment; the  parallel  between  the  two  cases  lies  in  their  complete 
irregularity,  but  this  in  itself  implies  a  close  resemblance.  One 
might  see  the  same  thing  in  a  crowded  market-place,  always  provided 
that  the  busthng  crowd  had  no  business  whatsoever.  In  like 
manner  Lucretius,  and  Epicurus  before  him,  watched  the  dust-motes 
quivering  in  the  beam,  and  saw  in  them  a  mimic  representation, 
rei  simulacrum  et  imago,  of  the  eternal  motions  of  the  atoms.  Again 
the  same  phenomenon  may  be  witnessed  under  the  microscope,  in 
a  drop  of  water  swarming  with  Paramoecia  or  such-Hke  Infusoria; 
and  here  the  analogy  has  been  put  to  a  numerical  test.  Following 
with  a  pencil  the  track  of  each  little  swimmer,  and  dotting  its  place 
every  few  seconds  (to  the  beat  of  a  metronome),  Karl  Przibram 
found  that  the  mean  successive  distances  from  a  common  base-hne 
obeyed  with  great  exactitude  the  "Einstein  formula,"  that  is  to 
say  the  particular  form  of  the  "law  of  chance"  which  is  apphcable 
to  the  case  of  the  Brownian  movement  f.  The  phenomenon  is  (of 
course)  merely  analogous,  and  by  no  means  identical  with  the 
Brownian  movement;  for  the  range  of  motion  of  the  little  active 
organisms,  whether  they  be  gnats  or  infusoria,  is  vastly  greater  than 
that  of  the  minute  particles  which  are  passive  under  bombardment ; 
nevertheless  Przibram  is  inclined  to  think  that  even  his  compara- 
tively large  infusoria  are  small  enough  for  the  molecular  bombard- 
ment to  be  a  stimulus,  even  though  not  the  actual  cause,  of  their 
irregular  and  interrupted  movements  {. 

*  Nevertheless  there  may  be  a  certain  amount  of  bias  or  direction  in  these 
seemingly  random  divagations:  cf.  J.  Brownlee,  Proc.  R.S.E.  xxxi,  p.  262, 
1910-11;  F.  H.  Edgeworth,  Metron,  i,  p.  75,  1920;  Lotka,  Elem.  of  Physical 
Biology,  1925,  p.  344. 

t  That  is  to  say,  the  mean  square  of  the  displacements  of  a  particle,  in  any 
direction,  is  proportional  to  the  interval  of  time.  Cf.  K.  Przibram,  Ueber  die 
ungeordnete  Bewegung  niederer  Tiere,  Pfliigefs  Archiv,  cliii,  pp.  401-405,  1913; 
Arch.  f.  Entw.  Mech.  xliii,  pp.  20-27,  1917. 

X  All  that  is  actually  proven  is  that  "pure  chance"  has  governed  the  movements 
of  the  little  organism.  Przibram  has  made  the  analogous  observation  that 
infusoria,  when  not  too  crowded  together,  spread  or  diffuse  through  an  aperture 
from  one  vessel  to  another  at  a  rate  very  closely  comparable  to  the  ordinary  laws 
of  molecular  diffusion. 


II]  OF  THE  EFFECTS  OF  SCALE  77 

George  Johnstone  Stoney,  the  remarkable  man  to  whom  we  owe 
the  name  and  concept  of  the  electron,  went  further  than  this;  for 
he  supposed  that  molecular  bombardment  might  be  the  source  of 
the  life-energy  of  the  bacteria.  He  conceived  the  swifter  moving 
molecules  to  dive  deep  into  the  minute  body  of  the  organism,  and 
this  in  turn  to  be  able  to  make  use  of  these  importations  of  energy*. 

We  draw  near  the  end  of  this  discussion.  We  found,  to  begin 
with,  that  "scale"  had  a  marked  eifect  on  physical  phenomena,  and 
that  increase  or  diminution  of  magnitude  migl^t  mean  a  complete 
change  of  statical  or  dynamical  equiUbrium.  In  the  end  we  begin 
to  see  that  there  are  discontinuities  in  the  scale,  defining  phases  in 
which  different  forces  predominate  and  different  conditions  prevail. 
Life  has  a  range  of  magnitude  narrow  indeed  compared  to  that  with 
which  physical  science  deals ;  but  it  is  wide  enough  to  include  three 
such  discrepant  conditions  as  those  in  which  a  man,  an  insect  and 
a  bacillus  have  their  being  and  play  their  several  roles.  Man  is 
ruled  by  gravitation,  and  rests  on  mother  earth.  A  water-beetle 
finds  the  surface  of  a  pool  a  matter  of  Ufe  and  death,  a  perilous 
entanglement  or  an  indispensable  support.  In  a  third  world, 
where  the  bacillus  Hves,  gravitation  is  forgotten,  and  the  viscosity 
of  the  Hquid,  the  resistance  defined  by  Stokes's  law,  the  molecular 
shocks  of  the  Brownian  movement,  doubtless  also  the  electric 
charges  of  the  ionised  medium,  make  up  the  physical  environment 
and  have  their  potent  and  immediate  influence  on  the  organism. 
The  predominant  factors  are  no  longer  those  of  our  scale ;  we  have 
come  to  the  edge  of  a  world  of  which  we  have  no  experience,  and 
where  all  our  preconceptions  must  be  recast. 

*  Phil.  Mag.  April  1890. 


CHAPTER  III, 

THE  RATE  OF  GROWTH 

When  we  study  magnitude  by  itself,  apart  from  the  gradual 
changes  to  which  it  may  be  subject,  we  are  deahng  with  a  something 
which  may  be  adequately  represented  by  a  number,  or  by  means 
of  a  Hne  of  definite  length ;  it  is  what  mathematicians  call  a  scalar 
phenomenon.  When  we  introduce  the  conception  of  change  of 
magnitude,  of  magnitude  which  varies  as  we  pass  from  one  point 
to  another  in  space,  or  from  one  instant  to  another  in  time,  our 
phenomenon  becomes  capable  of  representation  by  means  of  a  line 
of  which  we  define  both  the  length  and  the  direction;  it  is  (in  this 
particular  aspect)  what  is  called  a  vector  phenomenon. 

When  we  deal  with  magnitude  in  relation  to  the  dimensions  of  space, 
our  diagram  plots  magnitude  in  one  direction  against  magnitude  in 
another — length  against  height,  for  instance,  or  against  breadth ;  and 
the  result  is  what  we  call  a  picture  or  outhne,  or  (more  correctly) 
a  "plane  projection"  of  the  object.  In  other  words,  what  we  call 
Form  is  a  ratio  of  magnitudes*  referred  to  direction  in  space. 

When,  in  deahng  with  magnitude,  we  refer  its  variations  to 
successive  intervals  of  time  (or  when,  as  it  is  said,  we  equate  it  with 
time),  we  are  then  dealing  with  the  phenomenon  of  growth;  and 
it  is  evident  that  this  term  growth  has  wide  meanings.  For  growth 
may  be  positive  or  negative,  a  thing  may  grow  larger  or  smaller, 
greater  or  less;  and  by  extension  of  the  concrete  signification  of 
the  word  we  easily  arid  legitimately  apply  it  to  non-material  things, 
such  as  temperature,  and  say,  for  instance,  that  a  body  "grows" 
hot  or  cold.  When  in  a  two-dimensional  diagram  we  represent  a 
magnitude  (for  instance  length)  in  relation  to  time  (or  "plot"  length 
against  time,  as  the  phrase  is),  we  get  that  kind  of  vector  diagram 
which  is  known  as  a  "curve  of  growth."  We  see  that  the  pheno- 
menon which  we  are  studying  is  a  velocity  (whose  "dimensions"  are 
space/time,  or  L/T),  and  this  phenomenon  we  shall  speak  of,  simply, 
as  a  rate  of  growth. 

In  various  conventional  wa-ys  we  convert  a  two-dimensional  into 

*  In  Aristotelian  logic.  Form  is  a  quality.  None  the  less,  it  is  related  to  qiuirUity't 
and  we  jfind  the  Schoolmen  speaking  of  it  as  qualitas  circa  quantitatem. 


CH.  Ill]  OF  CHANGE  OF  MAGNITUDE  79 

a  three-dimensional  diagram.  We  do  so,  for  example,  when,  by 
means  of  the  geometrical  method  of  "perspective,"  we  represent 
upon  a  sheet  of  paper  the  length,  breadth  and  depth  of  an  object 
in  three-dimensional  space,  but  we  do  it  better  by  means  of  contour- 
Hnes  or  "isopleths."  By  contour-lines  superposed  upon  a  map  of 
a  country,  we  shew  its  hills  and  valleys;  and  by  contour-Unes  we 
may  shew  temperature,  rainfall,  population,  language,  or  any  other 
*' third  dimension"  related  to  the  two  dimensions  of  the  map.  Time 
is  always  impHcit,  in  so  far  as  each  map  refers  to  its  own  date  or 
epoch;  but  Time  as  a  dimension  can  only  be  substituted  for  one  of 
the  three  dimensions  already  there.  Thus  we  may  superpose  upon 
our  map  the  successive  outlines  of  the  coast  from  remote  antiquity, 
or  of  any  single  isotherm  or  isobar  from  day  to  day.  And  if  in  hke 
manner  we  superpose  on  one  another,  or  even  set  side  by  side,  the 
outhnes  of  a  growing  organism — for  instance  of  a  young  leaf  and 
an  old,  we  have  a  three-dimensional  diagram  which  is  a  partial 
representation  (limited  to  two  dimensions  of  space)  of  the  organism's 
gradual  change  of  form,  or  course  of  development;  in  such  a  case 
our  contours  may,  for  the  purposes  of  the  embryologist,  be  separated 
by  time-intervals  of  a  few  hours  or  days,  or,  for  the  palaeontologist, 
by  interspaces  of  unnumbered  and  innumerable  years*. 

Such  a  diagram  represents  in  two  of  its  three  dimensions  form, 
and  in  two  (or  three)  of  its  dimensions  growth,  and  we  see  how 
intimately  the  two  concepts  are  correlated  or  interrelated  to  one 
another.  In  short  it  is  obvious  that  the  form  of  an  organism  is 
determined  by  its  rate  of  growth  in  various  directions ;  hence  rate 
of  growth  deserves  to  be  studied  as  a  necessary  preliminary  to  the 
theoretical  study  of  form,  and  organic  form  itself  is  found, 
mathematically  speaking,  to  be  di  function  of  time'\. 

*  Sometimes  we  find  one  and  the  same  diagram  suffice,  whether  the  time-intervals 
be  great  or  small;  and  we  then  invoke  " Wolff's  law"  (or  Kielmeyer's),  and  assert 
that  the  life-history  of  the  individual  repeats,  or  recapitulates,  the  history  of  the 
race.  This  "recapitulation  theory"  was  alt-important  in  nineteenth -century 
embryology,  but  was  criticised  by  Adam  Sedgwick  {Q.J. M.S.  xxxvi,  p.  38,  1894) 
and  many  later  authors;  cf.  J.  Needham,  Chemical  Embryology,  1931,  pp.  1629-1647. 

t  Our  subject  is  one  of  Bacon's  "Instances  of  the  Course"  or  studies  wherein 
we  "measure  Nature  by  periods  of  Time."  In  Bacpji's  Catalogue  of  Particular 
Histories,  one  of  the  odd  hundred  histories  or  investigations  which,  he  foreshadows 
is  precisely  that  which  we  are  engaged  on,  viz.  a  "History  of  the  Growth  and 
Increase  of  the  Body,  in  the  whole  and  in  its  parts." 


80  THE  RATE  OF  GROWTH  [ch. 

At  the  same  time,  we  need  only  consider  this  large  part  of  our 
subject  somewhat  briefly.  Though  it  has  an  essential  bearing  on 
the  problems  of  morphology,  it  is  in  greater  degree  involved  with 
physiological  problems;  also,  the  statistical  or  numerical  aspect  of 
the  question  is  peculiarly  adapted  to  the  mathematical  study  of 
variation  and  correlation.  These  important  subjects  we  must  not 
neglect;  but  our  main  purpose  will  be  served  if  we  consider  the 
characteristics  of  a  rate  of  growth  in  a  few  illustrative  cases,  and 
recognise  that  this  rate  of  growth  is  a  very  important  specific 
property,  with  its  own  characteristic  value  in  this  organism  or  that, 
in  this  or  that  part  of  each  organism,  and  in  this  or  that  phase  of 
its  existence. 

The  statement  which  we  have  just  made  that  "the  form  of  an 
organism  is  determined  by  its  rate  of  growth  in  various  directions," 
is  one  which  calls  for  further  explanation  and  for  some  measure  of 
quahfication. 

Among  organic  forms  we  shall  have  many  an  occasion  to  see  that 
form  may  be  due  in  simple  cases  to  the  direct  action  of  certain 
molecular  forces,  among  which  surface-tension  plays  a  leading  part. 
Now  when  surface-tension  causes  (for  instance)  a  minute  semifluid 
organism  to  assume  a  spherical  form,  or  gives  to  a  film  of  protoplasm 
the  form  of  a  catenary  or  of  an  elastic  curve,  or  when  it  acts  in 
various  other  ways  productive  of  definite  contours — just  as  it  does 
in  the  making  of  a  drop,  a  splash  or  a  jet — this  is  a  process  of  con- 
formation very  diiferent  from  that  by  which  an  ordinary  plant  or 
animal  grows  into  its  specific  form.  In  both  cases  change  of  form 
is  brought  about  by  the  movement  of  portions  of  matter,  and  in 
both  cases  it  is  ultimately  due  to  the  action  of  molecular  forces; 
but  in  the  one  case  the  movements  of  the  particles  of  matter  lie  for 
the  most  part  within  molecular  range,  while  in  the  other  we  have 
to  deal  with  the  transference  of  portions  of  matter  into  the  system 
from  without,  and  from  one  widely  distant  part  of  the  organism  to 
another.  It  is  to  this  latter  class  of  phenomena  that  we  usually 
restrict  the  term  growth;  it  is  in  regard  to  them  that  we  are  in  a 
position  to  study  the  rate  of  action  in  different  directions  and  at 
different  times,  and  to  realise  that  it  is  on  such  differences  of  rate 
that  form  and  its  modifications  essentially  and  ultimately  depend. 


Ill]  OF  RATE  OF  ACTION  81 

The  difference  between  the  two  classes  of  phenomena  is  akin  to 
the  difference  between  the  forces  which  determine  the  form  of  a 
raindrop  and  those  which,  by  the  flowing  of  the  waters  and  the 
sculpturing  of  the  solid  earth,  have  brought  about  the  configuration 
of  a  river  or  a  hill ;  molecular  forces  are  paramount  in  the  one,  and 
wolar  forces  are  dominant  in  the  other. 

At  the  same  time,  it  is  true  that  all  changes  of  form,  inasmuch 
as  they  necessarily  involve  changes  of  actual  and  relative  magnitude, 
may  in  a  sense  be  looked  upon  as  phenomena  of  growth ;  and  it  is 
also  true,  since  the  movement  of  matter  must  always  involve  an 
element  of  time*,  that  in  all  cases  the  rate  of  growth  is  a  phenomenon 
to  be  considered.  Even  though  the  molecular  forces  which  play 
their  part  in  modifying  the  form  of  an  organism  exert  an  action 
which  is,  theoretically,  all  but  instantaneous,  that  action  is  apt  to 
be  dragged  out  to  an  appreciable  interval  of  time  by  reason  of 
viscosity  or  some  other  form  of  resistance  in  the  material.  From 
the  physical  or  physiological  point  of  view  the  rate  of  action  may  be 
well  worth  studying  even  in  such  cases  as  these;  for  example,  a 
study  of  the  rate  of  cell-division  in  a  segmenting  egg  may  teach  us 
something  about  the  w^ork  done,  and  the  various  energies  concerned. 
But  in  such  cases  the  action  is,  as  a  rule,  so  homogeneous,  and  the 
form  finally  attained  is  so  definite  and  so  little  dependent  on  the 
time  taken  to  effect  it,  that  the  specific  rate  of  change,  or  rate  of 
growth,  does  not  enter  into  the  morphological  problem. 

We  are  deahng  with  Form  in  a  very  concrete  way.  To  Aristotle 
it  was  a  metaphysical  concept;  to  us  it  is  a  quasi-mechanical  effect 
on    Matter    of   the    operation    of   chemico-physical    forces  f.     To 


*  Cf.  Aristotle,  Phys.  VI,  5,  235a,  11,  eirel  yap  awaaa  Kiurjffis  ev  XP^'^V*  kt\.;  he  had 
already  told  us  that  natural  science  deals  with  magnitude,  with  motion  and  with 
time:  ^<ttiu  ij  irepl  (pvaeus  eiriar-qtnj  irepl  fxiyedos  Kai  klvt^ctlv  koL  xP^^^^-  Hence 
omnis  velocitas  tempore  durat  became  a  scholastic  aphorism.  Bacon  emphasised,  in 
like  manner,  the  fact  that  "all  motion  or  natural  action  is  performed  in  time: 
some  more  quickly,  some  more  slowly,  but  all  in  periods  determined  and  fixed  in 
the  nature  of  things.  Even  those  actions  which  seem  to  be  performed  suddenly, 
and  (as  we  say)  in  the  twinkling  of  an  eye,  are  found  to  admit  of  degree  in 
respect  of  duration"  {Nov.  Organon,  xlvi).  That  infinitely  small  motions  take 
place  in  infinitely  small  intervals  of  time  is  the  concept  which  lies  at  the  root  of  the 
calculus.     But  there  is  another  side  to  the  story. 

t  Cf.  N.  K.  KoltzotF,  Physikalisch-chemische  Grundlage  der  Morphologie, 
Biol.  Centralbl.  1928,  pp.  345-369. 


82  THE  RATE  OF  GROWTH  [ch. 

Aristotle  its  Form  was  the  essence,  the  archetype,  the  very  "nature" 
of  a  thing,  and  Matter  and  Form  were  an  inseparable  duahty. 
Even  now,  when  we  divide  our  science  into  Physiology  and  Mor- 
phology, we  are  harking  back  to  the  old  Aristotehan  antithesis. 

To  sum  up,  we  may  lay  down  the  following  general  statements. 
The  form  of  organisms  is  a  phenomenon  to  be  referred  in  part  to 
the  direct  action  of  molecular  forces,  in  larger  part  to  a  more  complex 
and  slower  process,  indirectly  resulting  from  chemical,  osmotic  and 
other  forces,  by  which  material  is  introduced  into  the  organism 
and  transferred  from  one  part  of  it  to  another.  It  is  this  latter 
complex  phenomenon  which  we  usually  speak  of  as  "growth." 

Every  growing  organism;  and  every  part  of  such  a  growing 
organism,  has  its  own  specific  rate  of  growth,  referred  to  this  or 
that  particular  direction ;  and  it  is  by  the  ratio  between  these  rates 
in  different  directions  that  we  must  account  for  the  external  forms 
oiall  save  certain  very  minute  organisms.  This  ratio  may  sometimes 
be  of  a  simple  kind,  as  when  it  results  in  the  mathematically 
definable  outhne  of  a  shell,  or  the  smooth  curve  of  the  margin  of  a 
leaf.  It  may  sometimes  be  a  very  constant  ratio,  in  which  case  the 
organism  while  growing  in  bulk  suffers  httle  or  no  perceptible  change 
in  form;  but  such  constancy  seldom  endures  beyond  a  season,  and 
when  the  ratios  tend  to  alter,  then  we  have  the  phenomenon  of 
morphological  ''development,''  or  steady  and  persistent  alteration  of 
form. 

This  elementary  concept  of  Form,  as  determined  by  varying  rates 
of  Growth,  was  clearly  apprehended  by  the  mathematical  mind  of 
Haller — who  had  learned  his  mathematics  of  the  great  John 
Bemoulh,  as  the  latter  in  turn  had  learned  his  physiology  from  the 
writings  of  Borelh*  It  was  this  very  point,  the  apparently  un- 
limited extent  to  which,  in  the  development  of  the  chick,  inequalities 
of  growth  could  and  did  produce  changes  of  form  and  changes  of 
anatomical  structure,  that  led  Haller  to  surmise  that  the  process 
was  actually  without  Hmits,  and  that  all  development  was  but  an 
unfolding  or  "evolutio,"  in  which  no  part  came  into  being  which 

*  "Qua  in  re  Incomparabilis  Viri  Joh.  Alph.  Borelli  vestigiis  insistemus." 
Joh.  Bernoulli,  De  motu  musculorum,  1694. 


Ill]  THE  DOCTRINE  OF   PREFORMATION  83 

had  not  essentially  existed  before*.  In  short  the  celebrated  doctrine 
of  "preformation"  implied  on  the  one  hand  a  clear  recognition  of 
what  growth  can  do  throughout  the  several  stages  of  development, 
by  hastening  the  increase  in  size  of  one  part,  hindering  that  of 
another,  changing  their  relative  magnitudes  and  positions,  and  so 
altering  their  forms;  while  on  the  other  hand  it  betrayed  a  failure 
(inevitable  in  those  days)  to  recognise  the  essential  difference 
between  these  movements  of  masses  and  the  molecular  processes 
which  precede  and  accompany  them,  and  which  are  characteristic 
of  another  order  of  magnitude. 

The  general  connection,  between  growth  and  form  has  been 
recognised  by  other  writers  besides  Haller.  Such  a  connection  is 
imphcit  in  the  "proportional  diagrams"  by  which  Diirer  and  his 
brother-artists  illustrated  the  changes  in  form,  or  of  relative 
dimensions,  which  mark  the  child's  growth  to  boyhood  and  to 
manhood.  The  same  connection  was  recognised  by  the  early 
embryologists,  and  appears,  as  a  survival  of  the  doctrine  of  pre- 
formation, in  Pander's  I  study  of  the  development  of  the  chick. 
And  long  afterwards,  the  embryological  aspect  of  the  case  was 
emphasised  by  His  J,  who  pointed  out  that  the  foldings  of  the 
blastoderm,  by  which  the  neural  and  amniotic  folds  are  brought 
into  being,  were  the  resultant  of  unequal  rates  of  growth  in  what 
to  begin  with  was  a  uniform  layer  of  embryonic  tissue.  If  a  sheet 
of  paper  be  made  to  expand  here  and  contract  there,  as  by  moisture 
or  evaporation,  the  plane  surface  becomes  dimpled,  or  folded,  or 
buckled,  by  the  said  expansions  and  contractions;  and  the  dis- 
tortions to  which  the  surface  of  the  "germinal  disc"  is  subject  are, 
as  His  shewed  once  and  for  all,  precisely  analogous.     There  are 

*  Cf.  (e.g.)  Elem.  Physiologiae,  ed.  1766,  vm,  p.  114,  "Ducimur  autem  ad 
evolutionem  potissimum,  quando  a  perfecto  animale  retrorsum  progredimuis  et 
incrementonim  atque  mutationum  seriem  relegimus.  Ita  inveniemus  perfectum 
illud  animal  fuisse  imperfectius,  alterius  figurae  et  fabricae,  et  denique  rude  et 
informe :  et  tamen  idem  semper  animal  sub  iis  diversis  phasibus  fuisse,  quae  absque 
ullo  saltu  perpetuos  parvosque  per  gradus  cohaereant." 

t  Beitrdge  zur  Entwickelungsgeschichte  des  Huhnchens  im  Ei,  1817,  p.  40.  Roux 
ascribes  the  same  views  also  to  Von  Baer  and  to  R.  H.  Lotze  {Allgem.  Physiologie, 
1851,  p.  353). 

I  W.  His,  Unsere  Korperform,  und  das  physiologische  Problem  ihrer  Entstehung, 
1874.  See  also  Archiv  f.  Anatomie,  1894;  and  cf.  C.  B.  Davenport,  Processes  con- 
cerned in  Ontogeny,  Bull.  Mus.  Comp.  Anat,  xxvn,  1895;  also  G.  Dehnel  and 
Jan  Tur,  De  Embryonum  evolutionis  progress u  ineqvuli:  Kosmos  (Lwow),  lhi,  1928. 


84  THE  RATE  OF  GROWTH  [ch. 

certain  Nostoc-algae  in  which  unequal  growth,  ceasing  towards  the 
periphery  of  a  disc  and  increasing  here  and  there  within,  gives  rise 
to  folds  and  bucklings  curiously  Hke  those  of  our  own  ears:  which 
indeed  owe  their  shape  and  characteristic  folding  to  an  identical  or 
analogous  cause. 

An  experimental  demonstration  comparable  to  the  actual  case  is 
obtained  by  making  an  "artificial  blastoderm"  of  Uttle  pills  or 
pellets  of  dough,  which  are  caused  to  grow  at  varying  rates  by  the 
addition  of  varying  quantities  of  yeast.  Here,  as  Roux  is  careful 
to  point  out,*  it  is  not  only  the  growth  of  the  individual  cells,  but 
the  traction  exercised  on  one  another  through  their  mutual  inter- 
connections, which  brings  about  foldings,  wrinklings  and  other 
distortions  of  the  structure.  But  this  again,  or  such  as  this,  had  been 
in  Haller's  mind,  and  formed  an  essential  part  of  his  embry illogical 
doctrine.  For  he  has  no  sooner  treated  of  incrementum,  or  celeritas 
incrementi,  than  he  proceeds  to  deal  with  the  contributory  and 
complementary  phenomena  of  expansion,  traction  (adtractio)^  and 
pressure,  and  the  more  subtle  influences  which  he  denominates  vis 
derivationis  et  revulsionis%'.  these  latter  being  the  secondary  and 
correlated  effects  on  growth  in  one  part,  brought  about  by  such 
changes  as  are  produced,  for  instance  in  the  circulation,  by  the 
growth  of  another. 

We  have  to  do  with  growth,  with  exquisitely  graded  or  balanced 
growth,  and  with  forces  subtly  exerted  by  one  growing  part  upon 
another,  in  so  wonderful  a  piece  of  work  as  the  development  of  the 
eye :  as  its  primary  vesicle  expands  and  then  dimples  in,  as  the  lens 
appears  and  fits  into  place,  as  the  secondary  vesicle  closes  over  to 
form  iris  and  pupil,  and  in  all  the  rest  of  the  story. 

Let  us  admit  that,  on  the  physiological  side,  Haller's  or  His's 
methods  of  explanation  carry  us  but  a  little  way;  yet  even  this 
little  way  is  something  gained.     Nevertheless,  I  can  well  remember 

*  Roux,  Die  Entwickelungsmechanik,  1905,  p.  99. 

t  Op.  cit.  p.  302,  "Magnum  hoc  naturae  instrumentum,  etiam  in  corpore  animato 
evolvendo  potenter  operatur,  etc."  The  recurrent  laryngeal  nerve,  drawn  down 
as  its  arch  of  the  aorta  descends,  is  a  simple  instance  of  anatomical  traction.  The 
vitelline  and  omphalomesenteric  arteries  lead,  by  more  complicated  constraints 
and  tractions,  to  the  characteristic  loops  of  the  intestinal  blood  vessels,  and  of  the 
intestine  itself.     Cf.  G.  Enbom,  Lunds  Univ.  Arsskrift,  1939. 

X  Ibid.  p.  306,  "Subtiliora  ista,  et  aliquantum  hypothesi  mista,  tamen  magnam 
mihi  videntur  speciem  veri  habere." 


Ill]  OF  PHYSICS  AND  EMBRYOLOGY  85 

the  harsh  criticism  and  even  contempt  which  His's  doctrine  met 
with,  not  merely  on  the  ground  that  it  was  inadequate,  but  because 
such  an  explanation  was  deemed  wholly  inappropriate,  and  was 
utterly  disavowed*.  Oscar  Hertwig,  for  instance,  asserted  that,  in 
embryology,  when  we  find  one  embryonic  stage  preceding  another, 
the  existence  of  the  former  is,  for  the  embryologist,  an  all-sufficient 
"causal  explanation"  of  the  latter.  "We  consider  (he  says)  that 
we  are  studying  and  explaining  a  causal  relation  when  we  have 
demonstrated  that  the  gastrula  arises  by  invagination  of  a  blasto- 
sphere,  or  the  neural  canal  by  the  infolding  of  a  cell-plate  so  as  to 
constitute  a  tubef."  For  Hertwig,  then,  as  Roux  remarks,  the  task 
of  investigating  a  physical  mechanism  in  embryology — "der  Ziel  das 
Wirken  zu  erforschen" — has  no  existence  at  all.  For  Balfour  also, 
as  for  Hertwig,  the  mechanical  or  physical  aspect  of  organic  develop- 
ment had  little  or  no  attraction.  In  one  notable  instance,  Balfour 
himself  adduced  a  physical,  or  quasi-physical,  explanation  of  an 
organic  process,  when  he  referred  the  various  modes  of  segmentation 
of  an  ovum,  complete  or  partial,  equal  or  unequal  and  so  forth,  to 
the  varying  amount  or  varying  distribution  of  food-yolk  associated 
with  the  germinal  protoplasm  of  the  egg.  But  in  the  main,  like  all 
the  other  embryologists  of  his  day,  Balfour  was  engrossed  in  the 

*  Cf.  His,  On  the  Principles  of  Animal  Morphology,  Proc.  R.S.E.  xv, 
p.  294,  1888:  "My  own  attempts  to  introduce  some  elementary  mechanical  or 
physiolbgical  conceptions  into  embryology  have  not  generally  been  agreed  to  by 
morphologists.  To  one  it  seemed  ridiculous  to  speak  of  the  elasticity  of  the  germinal 
layers;  another  thought  that,  by  such  considerations,  we  'put  the  cart  before 
the  horse';  and  one  more  recent  author  states,  that  we  have  better  things  to  do 
in  embryology  than  to  discuss  tensions  of  germinal  layers  and  similar  questions, 
since  all  explanations  must  of  necessity  be  of  a  phylogenetic  nature.  This  opposition 
to  the  application  of  the  fundamental  principles  of  science  to  embryological  questions 
would  scarcely  be  intelligible  had  it  not  a  dogmatic  background.  No  other  explana- 
tion of  living  forms  is  allowed  than  heredity,  and  any  which  is  founded  on  another 
basis  must  be  rejected.. .  .To  think  that  heredity  will  build  organic  beings  without 
mechanical  means  is  a  piece  of  unscientific  mysticism."  Even  the  school  of 
Entwickelungsmechanik  showed  a  certain  reluctance,  or  extreme  caution,  in  speaking 
of  the  physical  forces  in  relation  to  embryology  or  physiology.  This  reluctant 
caution  is  well  exemplified  by  Martin  Heidenhain,  writing  on  "Formen  und  Krafte 
in  der  lebendigen  Natur"  in  Roux's  Vortrdge,  xxxir,  1923.  Speaking  of  "die 
Krafte  welche  die  Entwickelung  und  den  fertigen  Zustand  der  Formen  bedingen", 
he  says:  "letztere  kann  man  aber  nicht  auf  dem  Felde  der  Physik  suchen,  sondern 
nur  im  Umkreis  der  Lebendigen,  obwohl  anzunehmen  ist,  dass  diese  Krafte  spater 
einmal  '  analogienhaft '  nach  dem  Vorbilde  der  Physik  beschreibbar  sein  werden" 

t  0.  Hertwig,  Zeit-  und  Streitfragen  der  Biologie,  ii,  1897. 


86  THE  RATE  OF  GROWTH  [ch. 

problems  of  phylogeny,  and  he  expressly  defined  the  aims  of  com- 
parative embryology  (as  exemphfied  in  his  own  textbook)  as  being 
*' twofold:  (1)  to  form  a  basis  for  Phylogeny,  and  (2)  to  form  a  basis 
for  Organogeny,  or  the  origin  and  evolution  of  organs*." 

It  has  been  the  great  service  of  Roux  and  his  fellow-workers  of 
the  school  of  "Entwickelungsmechanik,"  and  of  many  other  students 
to  whose  work  we  shall  refer,  to  try,  as  His  tried,  to  import  into 
embryology,  wherever  possible,  the  simpler  concepts  of  physics,  to 
introduce  along  with  them  the  method  of  experiment,  and  to  refuse 
to  be  bound  by  the  narrow  hmitations  which  such  teaching  as  that 
of  Hertwig  would  of  necessity  impose  on  the  work  and  the  thought 
and  the  whole  philosophy  of  the  biologist. 

Before  we  pass  from  this  general  discussion  to  study  some  of  the 
particular  phenomena  of  growth,  let  m.e  give  an  illustration,  from 
Darwin,  of  a  point  of  view  which  is  in  marked  contrast  to  Haller's 
simple  but  essentially  mathematical  conception  of  Form. 

There  is  a  curious  passage  in  the  Origin  of  Species  "f,  where  Darwin 
is  discussing  the  leading  facts  of  enibryology,  and  in  particular 
Von  Baer's  "law  of  embryonic  resemblance."  Here  Darwin  says: 
"We  are  so  much  accustomed  to  see  a  difference  in  structure  between 
the  embryo  and  the  adult  that  we  are  tempted  to  look  at  this 
difference  as  in  some  necessary  manner  contingent  on  growth.  But 
there  is  no  reason  why,  for  instance,  the  wing  of  a  bat,  or  the  fin 
of  a  porpoise,  should  not  have  been  sketched  out  with  all  their  parts 
in  proper  proportion,  as  soon  as  any  part  became  visible."  After 
pointing  out  various  exceptions,  with  his  habitual  care,  Darwin 
proceeds  to  lay  down  two  general  principles,  viz.  "that  shght 
variations  generally  appear  at  a  not  very  early  period  of  Hfe,"  and 
secondly,  that  "at  whatever  age  a  variation  first  appears  in  the 
parent,  it  tends  to  reappear  at  a  corresponding  age  in  the  offspring." 
He  then  argues  that  it  is  with  nature  as  with  the  fancier,  who  does 
not  care  what  his  pigeons  look  Uke  in  the  embryo  so  long  as  the 
full-grown  bird  possesses  the  desired  quahties :  and  that  the  process 
of  selection  takes  place  when  the  birds  or  other  animals  are  nearly 

*  Treatise  on  Comparative  Embryology,  i,  p.  4,  1881. 

t  Ist  ed.  p.  444;  6th  ed.  p.  390.  The  student  should  not  fail  to  consult  the 
passage  in  question ;  for  there  is  always  a  risk  of  misunderstanding  or  misinterpreta- 
tion when  one  attempts  to  epitomise  Darwin's  carefully  condensed  arguments. 


Ill]  A  PASSAGE  IN  DARWIN  87 

grown  up — at  least  on  the  part  of  the  breeder,  and  presumably  in 
nature  as  a  general  rule.  The  illustration  of  these  principles  is  set 
forth  as  follows:  "Let  us  take  a  group  of  birds,  descended  from 
some  ancient  form  and  modified  through  natural  selection  for 
different  habits.  Then,  from  the  many  successive  variations  having 
supervened  iA  the  several  species  at  a  not  very  early  age,  and  having 
been  inherited  at  a  corresponding  age,  the  young  will  still  resemble 
each  other  much  more  closely  than  do  tl^e  adults — just  as  we  have 
seen  with  the  breeds  of  the  pigeon. . . .  Whatever  influence  long- 
continued  use  or  disuse  may  have  had  in  modifying  the  limbs  or 
other  parts  of  any  species,  this  will  chiefly  or  solely  have  affected 
it  when  nearly  mature,  when  it  was  compelled  to  use  its  full  powers 
to  gain  its  own  living ;  and  the  effects  thus  produced  will  have  been 
transmitted  to  the  offspring  at  a  corresponding  nearly  mature  age. 
Thus  the  young  will  not  be  modified,  or  will  be  modified  only  in  a 
shght  degree,  through  the  effects  of  the  increased  use  or  disuse*  of 
parts."  This  whole  argument  is  remarkable,  in  more  ways  than 
we  need  try  to  deal  with  here;  but  it  is  especially  remarkable  that 
Darwin  should  begin  by  casting  doubt  upon  the  broad  fact  that  a 
"difference  in  structure  between  the  embryo  and  the  adult"  is 
"in  some  necessary  matter  contingent  on  growth";  and  that  he 
should  see  no  reason  why  compUcated  structures  of  the  adult 
"should  not  have  been  sketched  out  with  all  their  parts  in  proper 
proportion,  as  soon  as  any  part  became  visible."  It  would  seem  to 
me  that  even  the  most  elementary  attention  to  form  in  its  relation 
to  growth  would  have  removed  most  of  Darwin's  difficulties  in  regard 
to  the  particular  phenomena  which  he  is  considering  here.  For 
these  phenomena  are  phenomena  of  form,  and  therefore  of  relative 
magnitude ;  and  the  magnitudes  in  question  are  attained  by  growth, 
proceeding  with  certain  specific  velocities,  and  lasting  for  certain 
long  periods  of  time.  And  it  seems  obvious  accordingly  that  in  any 
two  related  individuals  (whether  specifically  identical  or  not)  the 
differences  between  them  must  manifest  themselves  gradually,  and 
be  but  Httle  apparent  in  the  young.  It  is  for  the  same  simple 
reason  that  animals  which  are  of  very  different  sizes  when  adult 
differ  less  and  less  in  size  (as  well  as  form)  as  we  trace  them  back- 
wards to  their  early  stages. 

Though  we  study  the  visible  effects  of  varying  rates  of  growth 


88  THE  RATE  OF  GROWTH  [ch. 

throughout  wellnigh  all  the  problems  of  morphology,  it  is  not  very 
often  that  we  can  directly  measure  the  velocities  concerned.  But 
owing  to  the  obvious  importance  of  the  phenomenon  to  the  morpho- 
logist  we  must  make  shift  to  study  it  where  we  can,  even  though 
our  illustrative  cases  may  seem  sometimes  to  have  little  bearing 
on  the  morphological  problem*. 

In  a  simple  spherical  organism,  such  as  the  single  spherical  cell  of 
Protococcus  or  of  Orbulina,  growth  is  reduced  to  its  simplest  terms, 
and  indeed  becomes  so  simple  in  its  outward  manifestations  that  it 
loses  interest  to  the  morphologist.  The  rate  of  growth  is  measured 
by  the  rate  of  change  in  length  of  a  radius,  i.e.  F  =  {R'  —  R)IT,  and 
from  this  we  may  calculate,  as  already  indicated,  the  rate  in  terms 
of  surface  and  of  volume.  The  growing  body  remains  of  constant 
form,  by  the  symmetry  of  the  system;  because,  that  is  to  say,  on 
the  one  hand  the  pressure  exerted  by  the  growing  protoplasm  is 
exerted  equally  in  all  directions,  after  the  manner  of  a  hydrostatic 
pressure,  which  indeed  it  actually  is ;  while  on  the  other  hand  the 
"skin"  or  surface  layer  of  the  cell  is  sufficiently  homogeneous  to  exert 
an  approximately  uniform  resistance.  Under  these  simple  conditions, 
then,  the  rate  of  growth  is  uniform  in  all  directions,  and  does  not 
affect  the  form  of  the  organism." 

But  in  a  larger  or  a  more  complex  organism  the  study  of  growth, 
and  of  the  rate  of  growth,  presents  us  with  a  variety  of  problems, 
and  the  whole  phenomenon  (apart  from  its  physiological  interest) 
becomes  a  factor  of  great  morphological  importance.  We  no  longer 
find  that  growth  tends  to  be  uniform  in  all  directions,  nor  have  we 
any  reason  to  expect  it  should.  The  resistances  which  it  meets  with 
are  no  longer  uniform.  In  one  direction  but  not  in  others  it  will 
be  opposed  by  the  important  resistance  of  gravity;  within  the 
growing  system  itself  all  manner  of  structural  differences  come  into 
play,  and  set  up  unequal  resistances  to  growth  in  one  direction  or 
another.  At  the  same  time  the  actual  sources  of  growth,  the 
chemical  and  osmotic  forces  which  lead  to  the  intussusception  of 
new  matter,  are  not  uniformly  distributed;  one  tissue  or  one  organ 
may  well  increase  while  another  does  not;  a  set  of  bones,  their 
intervening  cartilages  and  their  surrounding  muscles,  may  all  be 

*  "In  omni  rerum  naturalium  historia  utile  est  mensuras  definiri  et  numeros,^^ 
Haller,  Elem.  Physiol,  ii,  p.  258,  1760.     Cf.  Hales,  Vegetable  Staticks,  Introduction. 


Ill]  ADOLPHE  QUETELET  89 

capable  of  very  different  rates  of  increment.  The  changes  of  form 
which  result  from  these  differences  in  rate  are  especially  manifested 
during  that  phase  of  life  when  growth  itself  is  rapid:  when  the 
organism^  as  we  say,  is  undergoing  its  development. 

When  growth  in  general  has  slowed  down,  the  differences  in  rate 
between  different  parts  of  the  organism  may  still  exist,  and  may  be 
made  manifest  by  careful  observation  and  measurement,  but  the 
resultant  change  of  form  is  less  apt  to  strike  the  eye.  Great  as  are 
the  differences  between  the  rates  of  growth  in  different  parts  of  a 
complex  organism,  the  marvel  is  that  the  ratios  between  them  are 
so  nicely  balanced  as  they  are,  and  so  capable  of  keeping  the  form 
of  the  growing  organism  all  but  unchanged  for  long  periods  of  time, 
or  of  slowly  changing  it  in  its  own  harmonious  way.  There  is  the 
nicest  possible  balance  of  forces  and  resistances  in  every  part  of 
the  complex  body;  and  when  this  normal  equilibrium  is  disturbed, 
then  we  get  abnormal  growth,  in  the  shape  of  tumours  and  exostoses, 
and  other  malformations  and  deformities  of  every  kind. 

The  rate  of  growth  in  man 
Man  will  serve  us  as  well  as  another  organism  for  our  first  illus- 
trations of  rate  of  growth,  nor  can  we  easily  find  another  which  we 
can  better  study  from  birth  to  the  utmost  Hmits  ofold  age.  Nor 
can  we  do  better  than  go  for  our  first  data  concerning  him  to 
Quetelet's  Essai  de  Physique  Sociale,  an  epoch-making  book  for  the 
biologist.  For  it  is  packed  with  information,  some  of  it  unsurpassed, 
in  regard  to  human  growth  and  form;  and  it  stands  out  as  the 
first  great  essay  in  which  social  statistics  and  organic  variation  are 
dealt  with  from  the  point  of  view  of  the  mathematical  theory  of 
probabilities.  How  on  the  one  hand  Quetelet  followed  Da  Vinci, 
Luca  Pacioli  and  Dlirer  in  studying  the  growth  and  proportions  of 
man :  and  how  on  the  other  he  simplified  and  extended  the  ideas  of 
James  Bernoulli,  of  d'Alembert,  Laplace,  Poisson  and  the  rest,  is 
another  and  a  vastly  interesting  story*. 

*  Quetelet,  Sur  V Homme,  ...,  ou  Essai  de  Physique  Sociale,  Bruxelles,  1835: 
trans.  Edinburgh,  1842;  a,\so  Instructions populaires sur  le  calcul des probabilites,  1828; 
Lettres.  .  .sur  la  theorie  des  probabilites  appliquee  aux  sciences  morales  et  politiques, 
1846 ;  and  Anthropometrie,  1871 .  For  an  account  of  his  life  and  writings,  see  Lottin's 
Quetelet,  statisticien  et  sociologue,  Louvain,  1912;  also  J.  M.  Keynes.  Treatise  on 
Probability,  1921. 


90 


THE  RATE  OF  GROWTH 


[CH. 

The  meaning  of  the  word  "statistics"  is  curiously  changed.  For 
Shakespeare  or  for  Milton  a  statist  meant  (so  Dr  Johnson  says) 
"a  pohtician,  a  statesman;  one  skilled  in  government."  The 
eighteenth-century  Statistical  Account  of  Scotland  was  a  description 
of  the  State  and  of  its  people,  its  wealth,  its.  agriculture  and  its  trade. 


Stature  and  weight*  of  man  (from  QuMeleVs  Belgian  data, 
Essai,  II,  pp.  23-43;  Anthropometric,  p.  346)  f 


Stature  in  metres 

A 

Weight  in  kgm. 

A. 

WjU 

xlOO 

A 

Age 

Male 

Female 

%  F/M 

Male 

Female 

%  F/M 

Male 

Female 

0 

0-50 

0-48 

96-0 

3-20 

2-91 

90-9 

2-56 

2-64 

1 

0-70 

0-69 

98-6 

1000 

9-30 

93-0 

2-92 

283 

2 

0-80 

0-78 

97-5 

1200 

11-40 

95-0 

2-35 

2-40 

3 

0-86 

0-85 

98-8 

13-21 

12-45 

94-2 

2-09 

203 

4 

0-93 

0-91 

97-6 

1507 

1418 

94-1 

1-84 

1-88 

6 

0-99 

0-97 

98-4 

16-70 

15-50 

92-8 

1-89 

1-69 

6 

105 

103 

98-6 

18-04 

16-74 

92-8 

1-56 

1-53 

1 

Ml 

MO 

98-6 

20-16 

18-45 

91-5 

1-48 

1-39 

8 

M7 

114 

97-3 

22-26 

19-82 

89-0 

1-39 

1-34 

9 

1-23 

1-20 

97-8 

24-09 

22-44 

93-2 

1-29 

1-30 

10 

1-28 

1-25 

97-3 

2612 

24-24 

92-8 

1-25 

1-24 

11 

1-33 

1-28 

96-1 

27-85 

26-25 

94-3 

1-18 

1-25 

12 

1-36 

1-33 

97-6 

31-00 

30-54 

98-5 

1-23 

1-38 

13 

1-40 

1-39 

98-8 

35-32 

34-65 

98-1 

1-29 

1-29 

14 

1-49 

1-45 

97-3 

40-50 

38-10 

94-1 

1-21 

1-25 

15 

1-56 

1-47 

94-6 

46-41 

41-30 

89-0 

1-22 

1-30 

16 

1-61 

1-52 

93-2 

53-39 

44-44 

83-2 

1-20 

1-32 

17 

1-67 

1-54 

92-5 

57-40 

49-08 

85-5 

1-23 

i-a4 

18 

1-70 

1-56 

91-9 

61-26 

5310 

86-.7 

1-24 

1-40 

19 

1-71 





63-32 





1-20 

. 

20 

1-71 

1-57 

91-8 

65-00 

54-46 

83-8 

1-30 

1-41 

25 

1-72 

1-58 

91-6 

68-29 

5508 

80-7 

1-39 

1-39 

30 

172 

1-58 

91-7 

68-90 

5514 

80-0 

1-35 

1-39 

40 

1-71 

1-56 

90-8 

68-81 

56-65 

82-3 

1-38 

1-49 

50 

1-67 

1-54 

91-8 

67-45 

58-45 

86-7 

1-45 

1-59 

60 

1-64 

1-52 

92-5 

65-50 

56:73 

86-6 

1-48 

1-61 

70 

1-62 

1-51 

93-3 

6303 

53-72 

85-2 

1-48 

1-58 

80 

1-61 

1-51 

93-4 

61-22 

51-52 

84-1 

1-46 

1-50 

This  is  what  Sir  Wilham  Petty  had  meant  in  the  seventeenth  century 
by  his  Political  Arithmetic,  and  what  Quetelet  meant  in  the  nine.teenth 
by  his  Physique  Sociale.  But  "statistics"  nowadays  are  counts  and 
measures  of  all  sorts  of  things;    and  statistical  science  arranges, 

*  The  figures  for  height  and  weight  given  in  my  first  edition  were  Quetelet's 
smoothed  or  adjusted  values.     I  have  gone  back  to  his  original  data. 

t  This  "almost  steady  growth,"  from  about  seven  years  old  to  eleven,  means 
that  the  curve  of  growth  is  a  nearly  straight  line  during  this  period:  a  result 
already  found  by  Elderton  for  Glasgow  children  {Biometrika,  x,  p.  293,  1914-15), 
by  Fessard  and  Laufer  in  Paris  {Nouvelles  Tables  de  Croissance,  1935,  p.  13),  etc. 


Ill] 


OF  STATISTICAL  METHODS 


91 


explains,  and  draws  deductions  from,  the  resulting  series  and  arrays 
of  numbers.  It  deals  with  simple  and  measurable  effects,  due  to 
complex  and  often  unknown  causes ;  and  when  experiment  is  not  at 
hand  to  disentangle  these  causes,  statistical  methods  may  still  do 
something  to  elucidate  them. 

Now  as  to  the  growth  of  man,  if  the  child  be  some  20  inches,  or 
say  50  cm.,  tall  at  birth,  and  the  man  some  six  feet,  or  180  cm., 
high  at  twenty,  we  may  say  that  his  average  rate  of  growth  had 


10  13 

Time  in  years 


20 


Fig.  4.     Curve  of  growth  in  man.   From  Quet^let's  Belgian  data. 
The  curve  H*  is  proportional  to  the  height  cubed. 

been  (180  -  50)/20  cm.,  or  6-5  cm.  per  annum.  But  we  well  know 
that  this  is  but  a  rough  preliminary  statement,  and  that  growth 
was  surely  quick  during  some  and  slow  during  other  of  those  twenty 
years;  we  must  learn  not  only  the  result  of  growth  but  the  course 
of  growth ;  we  must  study  it  in  its  continuity.  This  we  do,  in  the 
first  instance,  by  the  method  of  coordinates,  plotting  magnitude 
against  time.  We  measure  time  along  a  certain  axis  (x),  and  the 
magnitude  in  question  along  a  coordinate  axis  (y);  a  succession  of 
points  defines  the  magnitudes  reached  at  corresponding  epochs,  and 
these  points  constitute  a  ''curve  of  growth''  when  we  join  them 
together. 


92  THE  RATE  OF  GROWTH  [ch. 

Our  curve  of  growth,  whether  for  weight  or  stature,  has  a  definite 
form  or  characteristic  curvature:  this  being  a  sign  that  the  rate  of 
growth  is  not  always  the  same  but  changes  as  time  goes  on.  Such 
as  it  is,  the  curvature  alters  in  an  orderly  way;  so  that,  apart  from 
minor  and  "fortuitous"  irregularities,  our  curves  of  growth  tend  to 
be  smooth  curves.  And  the  fact  that  they  are  so  is  an  instance  of 
that  "principle  of  continuity"  which  is  the  foundation  of  all  physical 
and  natural  science. 

The  curve  of  growth  (Fig.  4)  for  length  or  stature  in  man  indicates 
a  rapid  increase  at  the  outset,  during  the  quick  growth  of  babyhood ; 
a  long  period  of  slower  but  almost  steady  growth  in  boyhood;  as 
a  rule  a  marked  quickening  in  his  early  teens,  when  the  boy  comes 
to  the  "growing  age";  and  a  gradual  arrest  of  growth  as  he  "comes 
to  his  full  height"  and  reaches  manhood.  If  we  carried  the  curve 
farther,  we  should  see  a  very  curious  thing.  We  should  see  that  a 
man's  full  stature  endures  but  for  a  spell;  long  before  fifty*  it  has 
begun  to  abate,  by  sixty  it  is  notably  lessened,  in  extreme  old  age 
the  old  man's  frame  is  shrunken  and  it  is  but  a  memory  that  "he 
once  was  tall";  the  dechne  sets  in  sooner  in  women  than  in  men, 
and  "a  httle  old  woman"  is  a  household  word.  We  have  seen, 
and  we  see  again,  that  growth  may  have  a  negative  value,  pointing 
towards  an  inevitable  end.  The  phenomenon  of  negative  growth 
extends  to  weight  also;  it  is  largely  chemical  in  origin;  the  meta- 
bolism of  the  body  is  impaired,  and  the  tissues  keep  pace  no  longer 
with  senile  wastage  and  decay. 

We  must  be  very  careful,  however,  how  we  interpret  such  a  Table 
as  this;  for  it  records  the  character  of  a  population,  and  we  are  apt 
to  read  in  it  the  life-history  of  the  individual.  The  two  things  are 
not  necessarily  the  same.  That  a  man  grows  less  as  he  grows  older 
all  old  men  know;  but  it  may  also  be  the  case,  and  our  Table  may 
indicate  it,  that  the  short  men  live  longer  than  the  tall. 

Our  curve  of  growth  is,  by  implication,  a  "time-energy"  diagram  f 
or  diagram  of  activity.  As  man  grows  he  is  absorbing  energy 
beyond  his  daily  needs,  and  accumulating  it  at  a  rate  depicted  in 

*  Dr  Johnson  was  not  far  wrong  in  saying  that  "life  declines  from  thirty-five"; 
though  the  Autocrat  of  the  Breakfast-table  declares,  like  Cicero,  that  "the  furnace 
is  in  full  blast  for  ten  years  longer". 

t  J.  Joly,  The  Abundance  of  Life,  1915  (1890),  p.  86. 


Ill]  OF  MAN'S  GROWTH  AND  STATURE  93 

our  curve ;  till  the  time  comes  when  he  accumulates  no  longer,  and 
is  constrained  to  draw  upon  his  dwindhng  store.  But  in  part,  tte 
slow  decline  in  stature  is  a  sign  of  the  unequal  contest  between  our 
bodily  powers  and  the  unchanging  force  of  gravity,  which  draws  us 
down  when  we  would  fain  rise  up*;  we  strive  against  it  all  our 
days,  in  every  movement  of  our  limbs,  in  every  beat  of  our  hearts. 
Gravity  makes  a  difference  to  a  man's  height,  and  no  slight  one, 
between  the  morning  and  the  evening ;  it  leaves  its  mark  in  sagging 
wrinkles,  drooping  mouth  and  hanging  breasts ;  it  is  the  indomitable 
force  which  defeats  us  in  the  end,  which  lays  us  on  our  death-bed 
and  lowers  us  to  the  grave  f.  But  the  grip  in  which  it  holds  us  is 
the  title  by  which  we  live;  were  it  not  for  gravity  one  man  might 
hurl  another  by  a  puff  of  his  breath  into  the  depths  of  space,  beyond 
recall  for  all  eternity  {. 

Side  by  side  with  the  curve  which  represents  growth  in  length, 
or  height  or  stature,  our  diagram  shews  the  corresponding  curve  of 
weight.  That  this  curve  is  of  a  different  shape  from  the  former  one 
is  accounted  for  in  the  main  (though  not  wholly)  by  the  fact — 
which  we  have  already  dealt  with — that  in  similar  bodies  volume, 
and  therefore  weight,  varies  as  the  cubes  of  the  linear  dimensions; 
and  drawing  a  third  curve  to  represent  the  cubes  of  the  corresponding 
heights,  it  now  resembles  the  curve  of  weight  pretty  closely,  but 
still  they  are  not  quite  the  same.  There  is  a  change  of  direction, 
or  "point  of  inflection,"  in  the  curve  of  weight  at  one  or  two  years 
old,  and  there  are  certain  other  features  in  our  curves  which  the 
scale  of  the  diagram  does  not  make  clear;  and  all  these  differences 
are  due  to  the  fact  that  the  child  is  changing  shape  as  he  grows, 
that    other   linear    dimensions    grow    somewhat   differently   from 

*  "Lou  pes,  mestre  de  tout  (Le  poids,  maitre  de  tout),  m^stre  senso  vergougno, 
Que  te  tirasso  en  bas  de  sa  brutalo  pougno."  J.  H.  Fabre,  Oubreto  prouven^alo, 
p.  61. 

t  The  continuity  of  the  phenomenon  of  growth,  and  the  natural  passage  from 
the  phase  of  increase  to  that  of  decrease  or  decay,  are  admirably  discussed  by 
Enriques,  in  La  Morte,  Rivista  di  Scienza,  1907,  and  in  Wachstum  und  seine 
analytische  Darstellung,  Biol.  Centralbl.  June,  1909.  Haller  [Elementa,  vii, 
p.  68)  recognised  decrementum  as  a  phase  of  growth,  not  less  important  (theoretically) 
than  incrementum ;  ''Hristis,  sed  copiosa,  haec  est  materies." 

X  Boscovich,  Theoria,  para.  552,  "Homo  hominem  arreptum  a  Tellure,  et 
utcumque  exigua  impulsum  vi  vel  uno  etiam  oris  flatu  impetitum,  ab  hominum 
omnium  commercio  in  infinitum  expelleret,  nunquam  per  totam  aeternitatem 
rediturum." 


94 


THE  RATE  OF  GROWTH 


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Ill]  OF  CURVES  OF  GROWTH  95 

length  or  stature,  and  in  short  that  infant,  boy  and  man  are  not 
similar  figures*.  The  change  of  form  seems  sHght  and  gradual,  but 
behind  it  he  other  and  more  complex  things.  Th6  changing  ratio 
between  height  and  weight  imphes  changes  in  the  child's  metabolism^ 
in  the  income  and  expenditure  of  the  body.  The  infant  stores  up 
fat,  and  the  active  child  "runs  it  off"  again;  at  four  years  old  or 
five,  bodily  metaboUsm  and  increase  of  weight  are  at  a  minimum; 
but  a  fresh  start  is  made,  a  new  "nutritional  period"  sets  in,  and  the 
small  schoolboy  grows  stout  and  sJ3rong|. 

Our  curve  of  growth  shews  at  successive  epochs  of  time  the  height 
or  weight  which  has  been  reached  by  then;  it  plots  changing 
magnitude  (y)  against  advancing  time  (x).  It  is  essentially  a 
cumulative  or  summation  curve;  it  sums  up  or  "integrates"  all  the 
successive  magnitudes  which  have  been  added  in  all  the  foregoing 
intervals  of  time.  Where  the  curve  is  steep  it  means  that  growth 
was  rapid,  and  when  growth  ceases  the  curve  becomes  a  horizontal 
line.  It  follows  that,  by  measuring  the  slope  or  steepness  of  our 
curve  of  growth  at  successive  epochs,  we  shall  obtain  a  picture  of 
the  successive  velocities  or  growth-rates. 

The  steepness  of  a  curve  is  measured  by  its  "gradient  J,"  or  we 
may  roughly  estimate  it  by  taking  for  equal  intervals  of  time 
(strictly  speaking,  for  each  infinitesimal  interval  of  time)  the  incre- 
ment added  during  that  interval;  and  this  amounts  in  practice  to 
taking  the  differences  between  the  values  given  for 'the  successive 
epochs,  or  ages,  which  we  have  begun  by  studying.  Plotting  these 
successive  differences  against  time,  we  obtain  a  curve  each  point  on 
which  represents  a  certain  rate  at  a  certain  time;  and  while  the 
former  curve  shewed  a  continuous  succession  of  varying  magnitudes, 
this  shews  a  succession  of  varying  velocities.  The  mathematician 
calls  it  a  curve  of  first  differences ;  we  may  call  it  a  curve  of  annual 
(or  other)  increments ;  but  we  shall  not  go  wrong  if  we  call  it  a  curve 
of  the  rate  (or  rates)  of  growth,  or  still  more  simply,  a  velocity-curve. 

*  According  to  Quetelet's  data,  man's  stature  is  multiplied  by  3-4  and  his  weight 
by  20-3,  between  birth  and  the  age  of  twenty-one.  But  the  cube  of  3-4  is  nearly 
40;  so  the  weight  at  birth  should  be  multiplied  forty  times  by  the  age  of 
twenty-one,  if  infant,  boy  and  man  were  similar  figures. 

t  Cf.  T.  W.  Adams  and  E.  P.  Poulton,  Heat  production  in  man,  Ouy's  Hospital 
Reports  (4),  xvn,  1937,  and  works  quoted  therein. 

I  That  is,  by  its  trigonometrical  tangent,  referred  to  the  base-line. 


96 


THE  RATE  OF  GROWTH 


[CH. 


We  have  now  obtained  two  different  but  closely  related  curves, 
based  on  the  selfsame  facts  or  observations,  and  illustrating  them 
in  different  ways.  One  is  the  inverse  of  the  other ;  one  is  the  integral 
and  one  the  differential  of  the  other;  and  each  makes  clear  to  the 
eye  phenomena  which  are  imphcit,  but  are  less  conspicuous,  in  the 
other.  We  are  using  mathematical  terms  to  describe  or  designate 
them;  but  these  "curves  of  growth"  are  more  comphcated  than 
the  curves  with  which  mathematicians  are  wont  to  deal.  In  our 
study  of  growth  we  may  well  hope  to  find  curves  simpler  than  these; 


20» 


3       5        7       9       II      13      15      17      19     21 
Age  in  years 
Fig.  5.     Annual  increments  of  growth  in  man.     From  Quetelet's  Belgian  data. 

but  in  the  successive  annual  increments  of  a  boy's  growth  (as  Fig.  5 
exhibits  them)  we  are  deahng  with  no  one  continuous  operation 
(such  as  a  mathematical  formula  might  define),  but  with  a  succession 
of  events,  changing  as  times  and  circumstances  change. 

Our  curve  of  increments,  or  of  first  differences,  for  man's  stature 
(Fig.  5)  is  based,  perforce,  on  annual  measurements,  and  growth 
alters  quickly  enough  at  certain  ages  to  make  annual  intervals  unduly 
long;  nevertheless  our  curve  shews  several  important  things.  It 
suffices  to  shew,  for  length  or  stature,  that  the  growth-rate  in  early 
infancy  is  such  as  is  never  afterwards  re-attained.  From  this  high 
early  velocity  the  rate  on  the  whole  falls  away,  until  growth  itself 


Ill] 


OF  HUMAN  STATURE 


97 


comes  to  an  end  * ;  but  it  does  so  subject  to  certain  important  changes 
and  interruptions,  which  are  much  the  same  whether  we  draw  them 
from  Quetelet's  Belgian  data,  or  from  the  British,  American  and 
other  statistics  of  later  writers.  The  curve  falls  fast  and  steadily 
during  the  first  couple  of  years  of  the  child's  hfe  (a).  It  runs  nearly 
level  during  early  boyhood,  from  four  or  five  years  old  to  nine  or 
ten  (6).  Then,  after  a  brief  but  unmistakable  period  of  depression! 
during  which  growth  slows  down  still  more  (c),  the  boy  enters  on 

Annual  increments  of  stature  and  of  weight  in  man 
{After  Quetelet;  see  Table,  p.  90) 


Stature 

A 

(cm.) 

Weight 
Male 

(kgm.) 

Age 

Male 

Female 

Fema 

0-  1 

20 

21 

6-8 

6-4 

1-  2 

10 

9 

2-0 

1-9 

2-  3 

6 

7 

1-2 

11 

3-  4 

7 

6 

1-9 

1-7 

4-  5 

6 

6 

1-6 

1-3 

5-  6 

6 

6 

1-3 

1-2 

6-  7 

6 

7 

2-1 

1-7 

7-  8 

6 

4 

21 

1-4 

8-  9 

6 

6 

1-8 

2-6 

9-10 

5 

5 

2-0 

1-8 

10-11 

5 

3 

1-7 

2-0 

11-12 

3 

5 

3-2 

4-3 

12-13 

4 

6 

4-3 

4-1 

13-14 

9 

6 

5-2 

3-5 

14-15 

7 

2 

5-9 

3-2 

15-16 

5 

3 

7-0 

2-1 

16-17 

6 

4 

40 

4-6 

17-18 

3 

2 

3-9 

40 

18-19 

1 

1 

21 

1-4 

19-20 

0 

0 

1-7 

— 

his  teens  and  begins  to  "grow  out  of  his  clothes";  it  is  his  "growing 
age",  and  comes  to  its  height  when  he  is  about  thirteen  or  fourteen 
years  old  (d).  The  lad  goes  on  growing  in  stature  for  some  years  more, 
but  the  rate  begins  to  fall  off  (e),  and  soon  does  so  with  great  rapidity. 
The  corresponding  curve  of  increments  in  weight  is  not  very 
different  from  that  for  stature,  but  such  differences  as  there  are 

*  As  Haller  observed  it  to  do  in  the  chick f  "Hoc  iterum  incrementum  miro 
ordine  distribuitur,  ut  in  principio  incubationis  maximum  est;  inde  perpetuo 
minuatur"  {Elementa  Pkysiologiae,  viii,  p.  294).  Or  as  Bichat  says,  "II  y  a 
surabondance  de  vie  dans  I'enfant"  {Sur  la  Vie  et  la  Mort,  p.  1). 

t  This  depression,  or  slowing  down  before  puberty,  seems  to  be  a  universal 
phenomenon,  common  to  all  races  of  men.  It  is  a  curious  thing  that  Quetelet's 
"adjusted  figures"  (which  I  used  in  my  first  edition)  all  but  smooth  out  of 
recognition  this  characteristic  feature  of  his  own  observations. 


98  THE  RATE  OF  GROWTH  [ch. 

between  them  are  significant  enough.  There  is  some  tendency  for 
growth  in  weight  to  fall  off  or  fluctuate  at  four  or  five  years  old, 
before  the  small  boy  goes  to  school ;  but  there  is,  or  should  be,  httle 
retardation  of  weight  when  growth  in  height  slows  down  before 
he  enters  on  his  teens*.  The  healthy  lad  puts  on  weight  again 
more  and  more  rapidly,  for  some  httle  while  after  gjowth  in  stature 
has  slowed  down;  and  normal  increase  of  weight  goes  on,  more 
slowly,  while  the  man  is  "fiUing  out,"  long  after  growth  in  stature 
has  come  to  an  end.  But  somewhere  about  thirty  he  begins  losing 
weight  a  httle ;  and  such  subsequent  slow  changes  as  men  commonly 
undergo  we  need  not  stop  to  deal  with. 

The  differences  in  stature  and  build  between  one  race  and  another 
are  in  hke  manner  a  question  of  growth-rate  in  the  main.  Let  us 
take  a  single  instance,  and  compare  the  annual  increments  of 
growth  in  Chinese  and  Enghsh  boys.  The  curves  are  much  the 
same  in  form,  but  differ  in  amplitude  and  phase.  The  Enghsh  boy 
is  growing  faster  all  the  while;  but  the  minimal  rate  and  the 
maximal  rate  come  later  by  a  year  or  more  than  in  the  Chinese 
curve  t  (Fig.  6). 

Quetelet  was  not  the  first  to  study  man's  growth  and  stature, 
nor  was  he  the  first  student  of  social  statistics  and  "demography." 
The  foundations  of  modern  vital  statistics  had  been  laid  by  Graunt 
and  Petty  in  the  seventeenth  century  J;  the  economists  developed 
the  subject  during  the  eighteenth §,  and  parts  of  it  were  studied 

♦  That  the  annual  increments  of  weight  in  boys  are  nearly  constant,  and  the 
curve  of  growth  nearly  a  straight  line  at  this  age,  especially  from  about  8  to  11, 
has  been  repeatedly  noticed.  Cf.  Elderton,  Glasgow  School-children,  Biometrika, 
X,  p.  283,  1914-15;  Fessard  and  others,  Croissance  des  Ecoliers  Pariaiens,  1934, 
p.  13.  But  careful  measurements  of  American  children,  by  Katherine  Simmons 
and  T.  Wingate  Todd,  shew  steadily  increasing  increments  from  four  years  old 
tiU  puberty  {Growth,  n,  pp.  93-133,  1938). 

t  For  copious  bibliography,  see  J.  Needham,  op.  cit.,  also  Gaston  Backmah,  Das 
Wachstum der  Korperlange  des Menschen,  K.  Sv.Vetensk.  Akad.  Hdlgr.  (3),  xiv,  1934. 

X  Cf.  John  Graunt's  Natural  and  Political  Observations. .  .upon  the  Bills  of 
Mortality,  London,  1662;  The  Economic  Writings  of  Sir  William  Petty,  ed.  by 
C.  H.  Hull,  2  vols,,  Cambridge,  Mass.,  1927.  Concerning  Graimt  and  Petty — two 
of  the  original  Fellows  of  the  Royal  Society — see  {int.  al.)  H.  Westergaard,  History 
of  Statistics,  1932,  and  L.  Hogben  (and  others).  Political  Arithmetic,  1938. 

§  Besides  the  many  works  of  the  economists,  cf.  J.  G.  Roederer,  Sermo  de 
pondere  et  longitudine  recens-natorum.  Comment.  Soe.  Beg.  Sci.  Oottingae,  m, 
1753;  J.  F.  G.  Dietz,  De  temporum  in  graviditate  et  partu  aestimatione,  Diss., 
Gottingen,  1757. 


Ill] 


OF  HUMAN  STATURE 


99 


7 

1         1         1         1         1 

1          1         1 

1 

6 

*x   N^ 

- 

5 

\        y^ 

\      \ 
\      \ 

a  4 

^^        '>            / 

\      \ 

a 

v         r 

6  ^ 

Chinese 

V 

2 

\ 

1 
0 

1        f,       1        1        1 

1      1      1 

1 

7 

k               9/,o                        12/,  3 

•6/l7 

I7/|8 

Age 

ig.  6.     Annual  increments  of  stature. 

From  Roberts 

(English) 

and  Appleton's  (Chinese)  data. 

•Date 
1760*61  ^62'63'64'65  W67^68*6970'7r72*73'74V5*76  1777 

|6'0 


0    1     2    3    4    5    6    7    8    9   10  n  12  13  14  15  16  17  18 

Age  in  years 

Fig.  7.     Curve  of  growth  of  a  French  boy  of  the  eighteenth  century. 

From  Scammon,  after  BufFon. 


100 


THE  RATE  OF  GROWTH 


CH. 


eagerly  in  the  early  nineteenth,  when  the  exhaustion  of  the  armies 
of  France  and  the  evils  of  factory  labour  in  England  drew  attention 
to  the  stature  and  physique  of  man  and  to  the  difference  between 
the  healthy  and  the  stunted  child*. 

A  friend  of  BufPon's,  the  Count  Philibert  Gueneau  de  Montbeillard, 
kept  careful  measurements  of  his  own  son;  and  Buifon  pubhshed 
these  in  1777,  in  a  supplementary  volume  of  the  Histoire  Naturelle'f. 
The  child  was  born  in  April  1759;  it  was  measured  every  six  months 


1 
-  \ 

1  1  I 

1  1  1 

1      '      ' 

.      1      .      . 

20 

A 

— 

_  \ 

- 

- 

d 

- 

- 

\" 

- 

- 

A 

- 

10 

1 

1  1  1 

b        . 

1      1 

A 

1     1     1     1 

1     1 

10 


20 


Fig.  8.     Annual  increments  of  stature  of  the  said  French  boy. 

for  seventeen  years,  and  the  record  gives  a  curve  of  great  interest 
and  beauty  (Fig.  7).  There  are  two  ways  of  studying  such  a 
phenomenon — the  statistical  method  based  on  large  numbers,  and 
the  careful  study  of  the  individual  case;  the  curve  of  growth  of  this 
one  French  child  is  to  all  intents  and  purposes  identical,  save  that 
the  boy  was  throughout  a  trifle  taller,  with  the  mean  curve  yielded 
by  a  recent  study  of  forty-four  thousand  Uttle  Parisians  {. 

In  young  Montbeillard's  case  the  "curve  of  first  differences,"  or 
of  the  successive  annual  increments  of  stature  (Fig.  8),  is  clear  and 
beautiful.    It  shews  (a)  the  rapid,  but  rapidly  diminishing,  rate  of 

*  Cf.  M.  Hargenvilliers,  Recherches . . . sur . .  .le  recrutement  de  Varmie  en  France, 
1817;  J.  W.  Cowell,  Measurements  of  children  in  Manchester  and  Stockport, 
Factory  Reports,  i;  and  works  referred  to  by  Quetelet. 

t  See  Richard  E.  Scammon,  The  first  seriatim  study  of  human  growth,  Amer. 
Journ.  of  Physical  Anthropology,  x,  pp.  329-336,  1927. 

J  MM.  Variot  et  Chaumet,- Tables  de  croissance,  dressees. .  .d'apr^s  les  mensura- 
tions de  44,000  enfants  parisiens,  Bull,  et  M4m.  Soc.  d'Anthropologie,  iii,  p.  55, 
1906. 


Ill]  OF  GROWTH  IN  MAN  AND  WOMAN  101 

growth  in  infancy ;  (6)  the  steady  growth  in  early  boyhood ;  (c)  the 
period  of  retardation  which  precedes,  and  (d)  the  rapid  growth  which 
accompanies,  puberty. 

Buifon,  with  his  usual  wisdom,  adds  some  remarks  of  his  own, 
which  include  two  notable  discoveries.  He  had  observed  that  a 
man's  stature  is  measurably  diminished  by  fatigue,  and  the  loss  soon 
made  up  for  in  repose;  long  afterwards  Quetelet  said,  to  the  same 
effect,  "le  lit  est  favorable  a  la  croissance,  et  le  matin  un  homme  est 
un  peu  plus  grand  que  le  soir."  Buff  on  asked  whether  growth 
varied  with  the  seasons,  and  Montbeillard's  data  gave  him  his  reply. 
Growth  was  quicker  from  April  to  October  than  during  the  rest  of 
the  year:  shewing  that  "la  chaleur,  qui  agit  generalement  sur  le 
developpement  de  tous  les  etres  organisees,  influe  considerablement 
sur  I'accroissement  du  corps  humain."  Between  five  years  old  and 
ten,  the  child  grew  seven  inches  during  the  five  summers,  but  during 
the  five  winters  only  four;  there  was  a  Hke  difference  again,  though 
not  so  great,  while  the  boy  was  growing  quickly  in  his  teens;  but 
there  were  no  seasonal  differences  at  all  from  birth  to  five  years  old, 
when  the  child  was  doubtless  sheltered  from  both  heat  and  cold*. 

On  rate  of  growth  in  man  and  ivoman 

That  growth  follows  a  different  course  in  boyhood  and  in  girlhood 
is  a  matter  of  common  knowledge;  but  differences  in  the  curves  "of 
growth  are  not  very  apparent  on  the  scale  of  our  diagrams.  They 
are  better  seen  in  the  annual  increments,  or  first  differences;  and 
we  may  further  simplify  the  comparison  by  representing  the  girl's 
weight  or  stature  as  a  percentage  of  the  boy's. 

Taking  weight  to  begin  with  (Fig.  9),  the  girl's  growth-rate  is 
steady  in  childhood,  from  two  or  three  to  six  or  seven  years  old, 

*  Growth-rates  based  on  the  continuqus  study  of  a  single  individual  are  rare; 
we  depend  mostly  on  average  measurements  of  many  individuals  grouped  according 
to  their  average  age.  That  this  is  a  sound  method  we  take  for  granted,  but  we  may 
lose  by  it  as  well  as  gain.  (See  above,  p.  92.)  The  chief  epochs  of  growth,  the  chief 
singularities  of  the  curve,  will  come  out  much  the  same  in  the  individual  and  in  the 
average  curve.  But  if  the  individual  curves  be  skew,  averaging  them  will  tend  to 
smooth  the  skewnesa  away;  and,  more  curiously,  if  they  be  all  more  or  less  diverse, 
though  all  symmetrical,  a  certain  skewness  will  tend  to  develop  in  the  composite  or 
average  curve.  Cf.  Margaret  Merrill,  The  relationship  of  individual  to  average 
growth,  Human  Biology,  iii.  pp.  37-70,  1931. 


102 


THE  RATE  OF  GROWTH 


[CH. 


J \ I L 


J \ L 


0         2        4         6         8        10       12       14        16       18  19 
Age 
Fig.  9.    Annual  increase  in  weight  of  Belgian  boys  and  girls. 
From  Quetelet's  data.    (Smoothed  curves.) 


10  15 

Age 

Fig.  10.    Percentage  ratio  of  female  weight  and  stature  to  male. 
From  Quetelet's  Belgian  data. 


23 


Ill] 


OF  GROWTH  IN  MAN  AND  WOMAN 


103 


just  as  is  the  boy's;  but  her  curve  stands  on  a  lower  level,  for  the 
little  maid  is  putting  on  less  weight  than  the  boy  (b).  Later  on, 
her  rate  accelerates  (c)  sooner  than  does  his,  but  it  neyer  rises  quite 
so  high  {d).  After  a  first  maximum  at  eleven  or  twelve  her  rate  of 
growth  slows  down  a  Httle,  then  rises  to  a  second  maximum  when 


112 

110 

108 

106 

104 

102 

100 

98 

96 

94 

92 

90 


Matle  weight  /   \y  / 


1. \ L 


J I L 


1  3         5         7         9         11        13 


Fig.  i: 


Relative  weight  of  American  boys  and  girls. 
From  Simmons  and  Todd's  data. 


she  is  sixteen  or  seventeen,  after  the  boy's  phase  of  quickest  growth 
is  over  and  done.  This  second  spurt  of  growth,  this  increase  of 
vigour  and  of  weight  in  the  girl  of  seventeen  or  eighteen,  Quetelet's 
figures  indicate  and  common  observation  confirms.  Last  of  all, 
while  men  stop  adding  to  their  weight  about  the  age  of  thirty  or 
before,  this  does  not  happen  to  women.  They  increase  in  weight, 
though  slowly,  till  much  later  on:  until  there  comes  a  final  phase, 
in  both  sexes  ahke,  when  weight  and  height  and  strength  decline 
together. 


104 


THE  RATE  OF  GROWTH 


[CH. 


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Ill 


OF  GROWTH  IN  CHILDHOOD 


105 


Stature  and  weight  of  American  children  (Ohio) 
(From  Katherine  Simmons  and  T.  Wingate  Todd's  data) 


Stature  (cm.) 

%  ratio 

Weight  (lbs.) 
Boys            Girls 

Age 

Boys 

Girls 

%  rati 

3  months 

61-3 

59-3 

96-7 

14-4 

13-1 

91-1 

1  year 

761 

.  74-2 

97-5 

23-9 

21-9 

91-8 

2     „ 

87-4 

86-2 

98-6 

29-1 

27-5 

94-7 

3    „ 

96-2 

95-5 

99-3 

33-5 

32-5 

96-9 

4    „ 

103-9 

103-2 

99-3 

38-4 

37-1 

96-7 

o     „ 

110-9 

110-3 

99-4 

43-2 

42-3 

98-1 

6     „ 

117-2 

117-4 

1001 

48-5 

48-6 

100-0 

7     „ 

123-9 

123-2 

99-4 

54-7 

54-0 

98-8 

8     „ 

130-1 

129-3 

99-4 

62-2 

61-5 

98-7 

9     „ 

136-0 

135-7 

99-7 

69-5 

70-9 

102-0 

10     „ 

141-4 

140-8 

99-6 

78-5 

77-6 

98-8 

11     » 

146-5 

147-8 

100-7 

86-5 

87-0 

100-6 

12     „ 

151-1 

155-3 

102-8 

92-7 

102-7 

110-7 

13    „ 

156-7 

159-9 

102-0 

102-8 

114-6 

111-4 

Mean  of  observed  incremeyits  of  stature  and  weight  of 
American  children 

Increment  of  stature  (mm.)         Increment  of  weight  (lbs.) 


'' 

% 

r 

/o 

Age 

Boys 

Girls 

ratio 

Boys 

Girls 

ratio 

3  m 

.  -  1  yr.           150-4 

150-1 

99-8 

9-32 

8-07 

93-8 

1  yr.  -  2  ", 

123-9 

132-0 

106-5 

4-97 

5-56 

1120 

2   , 

-  3   , 

88-1 

90-0 

102-2 

4-01 

4-18 

104-3 

3   , 

-  4   , 

73-9 

79-1 

106-9 

4-13 

4-46 

108-0 

4   , 

-  5    , 

69-4 

72-2 

104-0 

4-60 

4-55 

99-8 

5   , 

-  6   , 

67-0 

68-0 

101-5 

4-51 

5-08 

112-8 

6   , 

-  7    , 

64-1 

62-6 

97-6 

5-57 

5-40 

96-9 

7   , 

-  8   , 

61-2 

57-8 

94-4 

6-70 

6-65 

99-4 

8   , 

—9   , 

55-7 

60-1 

108-0 

6-64 

7-38 

1111 

9   , 

-10   , 

54-9 

57-7 

105-1 

7-92 

8-12 

104-8 

10   , 

-11    , 

51-9 

61-3 

118-2 

8-81 

9-58 

108-7 

11    , 

-12   , 

53-2 

66-9 

125-6 

9-54 

11-98 

1331 

12   , 

-13   , 

61-0 

55-1 

89-0 

10-90 

10-29 

94-7 

These  differences  between  the  two  sexes,  which  are  essentially 
phase-differences,  cause  the  ratio  between  their  weights  to  fluctuate 
in  a  somewhat  comphcated  way  (Figs.  10,  11).  At  birth  the  baby 
girl's  weight  is  about  nine-tenths  of  the  boy's.  She  gains  on  him  for 
a  year  or  two,  then  falls  behind  again ;  from  seven  or  eight  onwards 
she  gains  rapidly,  and  the  girl  of  twelve  or  thirteen  is  very  httle 
lighter  than  the  boy;  indeed  in  certain  American  statistics  she  is 
by  a  good  deal  the  heavier  of  the  two.     In  their  teens  the  boy  gains 


106 


THE  RATE  OF  GROWTH 


[CH. 


steadily,  and  the  lad  of  sixteen  is  some  15  per  cent,  heavier  than 
the  lass.  The  disparity  tends  to  diminish  for  a  while,  when  the 
maid  of  seventeen  has  her  second  spurt  of  growth ;  but  it  increases 
again,  though  slowly,  until  at  five-and-twenty  the  young  woman  is 
no  more  than  four-fifths  the  weight  of  the  man.  During  middle  life 
she  gains  on  him,  and  at  sixty  the  difference  stands  at  some  12 


1 1 \ r 


J i \ \ \ \ L 


0       2       4       6       8        10      12      14      16      18 

Age 

Fig.  12.     Annual  increments  of  stature,  in  boys  and  girls. 
From  Quetelet's  data.     (Smoothed  curves.) 


per  cent.,  not  far  from  the  mean  for  all  ages;   but  the  old  woman 
shrinks  and  dwindles,  and  the  difference  tends  to  increase  again. 

The  rate  of  increase  of  stature,  like  stature  itself,  differs  notably 
in  the  two  sexes,  and  the  differences,  as  in  the  case  of  weight,  are 
mostly  a  question  of  phase  (Fig.  12).  The  httle  girl  is  adding  rather 
more  to  her  stature  than  the  boy  at  four  years  old*,  but  she  grows 

*  This  early  spurt  of  growth  in  the  girl  is  shewn  in  Enghsh,  French  and  American 
observations,  but  not  in  Quetelet's. 


Ill 


OF  GROWTH  IN  CHILDHOOD 


107 


slower  than  he  does  for  a  few  years  thereafter  (6).  At  ten  years  old 
the  girl's  growth-rate  begins  to  rise  (c),  a  full  year  before  the  boy's; 
at  twelve  or  thirteen  the  rate  is  much  ahke  for  both,  but  it  has 
reached  its  maximum  for  the  girl.  The  boys'  rate  goes  on  rising, 
and  at  fourteen  or  fifteen  they  are  growing  twice  as  fast  as  the  girls. 
So  much  for  the  annual  increments,  as  a  rough  measure  of  the  rates 
of  growth.  In  actual  stature  the  baby  girl  is  some  2  or  3  per  cent, 
below  the  boy  at  birth;   she  makes  up  the  difference,  and  there  is 


7         9 
Age 
Fig.  13.     Ratio  of  female  stature  to  male. 


II        13        15 


From  Simmons  and  Todd's  data. 


From  R.  M.  Fleming's  data. 


good  evidence  to  shew  that  she  is  by  a  very  little  the  taller  for  a 
while,  at  about  five  years  old  or  six.  At  twelve  or  thirteen  she  is 
very  generally  the  taller  of  the  two,  and  we  call  it  her  "gawky 
age"  (Fig.  13). 

Man  and  woman  differ  in  length  of  life,  just  3,s  they  do  in  weight 
and  stature.  More  baby  boys  are  born  than  girls  by  nearly  5  peF 
cent.     The  numbers  draw  towards  equality  in  their  teens;    after 


108  THE  RATE  OF  GROWTH  '  [ch. 

twenty  the  women  begin  to  outnumber  the  men,  and  at  eighty-five 
there  are  twice  as  many  women  as  men  left  in  the  world*. 

Men  have  pondered  over  the  likeness  and  the  unhkeness  between 
the  short  lifetimes  and  the  long;  and  some  take  it  to  be  fallacious 
to  measure  all  alike  by  the  common  timepiece  of  the  sun.  Life, 
they  say,  has  a  varying  time-scale  of  its  own ;  and  by  this  modulus 
the  sparrow  hves  as  long  as  the  eagle  and  the  day-fly  as  the  manf. 
The  time-scale  of  the  living  has  in  each  case  so  strange  a  property 
of  logarithmic  decrement  that  our  days  and  years  are  long  in 
childhood,  but  an  old  man's  minutes  hasten  to  their  end. 

On  pre-natal  and  post-natal  growth 

The  rates  of  growth  which  we  have  so  far  studied  are  based  on 
annual  increments,  or  "first  differences"  between  yearly  determina- 
tions of  magnitude.  The  first  increment  indicates  the  mean  rate  of 
growth  during  the  first  year  of  the  infant's  life,  or  (on  a  further 
assumption)  the  mean  rate  at  the  mean  epoch  of  six  months  old; 
there  is  a  gap  between  that  epoch  and  the  epoch  of  birth,  of  which 
we  have  learned  nothing;  we  do  not  yet  know  whether  the  very 
high  rate  shewn  within  the  first  year  goes  on  rising,  or  tends  to  fall, 
as  the  date  of  birth  is  approached.  We  are  accustomed  to  inter- 
polate freely,  and  on  the  whole  safely,  between  known  points  on 
a  curve:  "si  timide  que  Ton  soit,  il  faut  bien  que  Ton  interpole," 
says  Henri  Poincare;  but  it  is  much  less  safe  and  seldom  justifiable 
(at  least  until  we  understand  the  physical  principle  involved  and 
its  mathematical  expression)  to  "extrapolate"  beyond  the  limits  of 
our  observations. 

We  must  look  for  more  detailed  observations,  and  we  may  learn 
much  to  begin  with  from  certain  old  tables  of  Russow's  J,  who  gives 

*  Cf.  F.  E.  A.  Crew's  Presidential  Address  to  Section  D  of  the  British  Association, 
1937. 

+  Cf.  Gaston  Backman,  Die  organische  Zeit,  Lunds  Universitets  Arsskrift,  xxxv, 
Nr.  7,  1939. 

J  Quoted  in  Vierordt's  Anatomische . . . Daten  und  Tabellen,  1906,  p.  13.  See 
also,  among  many  others,  Camerer's  data,  in  Pfaundler  and  Schlossman's  Hdb.  d. 
Kinderheilkunde,  i,  pp.  49,  424,  1908;  Variot,  op.  cit.;  for  pre-natal  growth,  R.  E. 
Scammon  and  L.  A.  Calkins,  Growth  in  the  Foetal  Period,  Minneapolis,  1929.  Also, 
on  this  and  many  other  matters,  E.  Faure-Fremiet,  La  cinetique  du  developpement, 
Paris,  1925;  and,  not  least,  J.  Needham,  Chemical  EinJbryology,  1931. 


in] 


OF  GROWTH  IN  CHILDHOOD 


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110 


THE  RATE  OF  GROWTH 


[CH. 


the  stature  of  the  infant,  month  by  month,  during  the  first  year  of 
its  hfe,  as  follows : 

Mean  growth  of  an  infant,  in  its  first  twelve-month 
(After  Russow) 


Age  (months) 

0       12       3       4 

5       6       7       8 

9 

10     11 

12 

Length  (cm.) 
Monthly  incre- 
ment (cm.) 

50     54     58     60     62 
—      4      4      2      2 

64     65     66     67-5 
2       111-5 

68 
0-5 

69     70-5 
1       1-5 

72 
1-5 

From  these  data  of  Russow's  for  German  children,  rough  as 
indeed  they  are,  from  Variot's  for  little  Parisians  (Fig.  14),  and  from 


3  10  12 

Age  in  months 

Fig.  14.     Growth  of  Parisian  children  (boys)  from  birth  to  twelve  months  old. 
From  G.  Variot's  data;  Russow's  German  data  are  also  shewn,  by   x  x  x  . 

many  more,  we  see  that  the  rate  of  growth  rises  steadily  and  even 
rapidly  as  we  pass  backwards  towards  the  date  of  birth.  It  is  never 
anything  like  so  great  again.  It  is  an  impressive  demonstration 
of  the  dynamic  potentiahty,  of  the  store  of  energy,  in  the  newborn 
child. 

But  birth  itself  is  but  an  incident,  an  inconstant  epoch,  in  the 
life  and  growth  of  a  viviparous  animal.     The  foal  and  the  lamb 


Ill]  OF  PRENATAL  GROWTH  111 

are  born  later  than  a  man-child;  the  puppy  and  the  kitten  are  born 
easier,  and  in  more  helpless  case  than  ours;  the  mouse  comes  into 
the  world  still  earher  and  more  inchoate,  so  much  so  that  even  the 
little  marsupial  is  scarcely  more  embryonic  and  unformed*.  We 
must  take  account,  so  far  as  each  case  permits,  of  pre-natal  or  intra- 
uterine growth,  if  we  are  to  study  the  curve  of  growth  in  its  entirety. 
According  to  Hisf,  the  following  are  the  mean  lengths  from  month 
to  month  of  the  imborn  child : 

Months  01  23456789  10 

(Birth) 

Length  (mm.)   0   7-5  40    84   162  275  352  402  443  472  490) 

500) 

Increment  per     —     7-5      32-5      44        78      113        77      '50       41        29        18) 
month  (mm.)  28) 

These  data  hnk  on  very  well  to  those  of  Russow,  which  we  have 
just  considered;  and  (though  His's  measurements  for  the  pre- 
natal months  are  more  detailed  than  are  those  of  Russow  for  the 
first  year  of  post-natal  hfe)  we  may  draw  a  continuous  curve  of 
growth  (Fig.  15)  and  of  increments  of  growth  (Fig.  16)  for  the 
combined  periods.  It  will  be  seen  at  once  that  there  is  a  "point 
of  inflection"  somewhere  about  the  fifth  month  of  intra-uterine  life; 
up  to  that  date  growth  proceeds  with  a  continually  increasing 
velocity.  After  that  date,  though  growth  is  still  rapid,  its  velocity 
tends  to  fall  away;  the  curve,  while  still  ascending,  is  becoming 
an  S-shaped  curve  (Fig.  15).  There  is  a  shght  break  between  our 
two  sets  of  statistics  at  the  date  of  birth,  an  epoch  regarding  which 
we  should  like  to  have  precise  and  continuous  information.  But 
we  can  see  that  there  is  undoubtedly  a  certain  shght  arrest  of  growth, 
or  diminution  of  the  rate  of  growth,  about  this  epoch ;  the  sudden 
change  of  nurture  has  its  inevitable  effect,  but  this  shght  tem- 

*  It  is  part  of  the  story,  though  Ijy  no  mean?  all,  that  (as  Minot  says)  the  larger 
the  litter  the  sooner  does  birth  take  place.  That  the  day-old  foal  or  fawn  can  keep 
pace  with  their  galloping  dams  is  very  remarkable;  it  is  usually  explained 
teleologically,  as  a  provision  of  Nature,  on  which  their  safety  and  their  survival 
depend.  But  the  fact  that  they  come  one  at  a  birth  has  at  least  something  to  do 
with  their  comparative  maturity. 

t  Unsere  Korperform  und  das  physiologische  Problem  ihrer  Entstehung,  Leipzig,  1874. 
On  growth  in  weight  of  the  human  embryo,  see  C.  M.  Jackson,  Amer.  Journ.  Anat. 
XVII,  p.  118,  1909;  also  J.  Needham,  op.  cit.  pp.  379-383. 


112 


THE  RATE  OF  GROWTH 


[CH. 


u 

crns 

70 

- 

^^ 

60 

- 

^^y^^^'^^ 

50 

/ 

40 

/ 

- 

30 

1 

Birth 

20 
10 

Lr-r        ,           ,.',..           1 

1 

8       10      12       14      16       18      20    22 

months 

Fig.  15.     Curve  of  growth  (in  length  or  stature)  of  child,  before  and  after  birth. 
From  His  and  Russow's  data. 


8         10        12        14 


18      20    22 
montha 


Fig.  16.     Mean  monthly  increments  of  length  or  stature  of  child,  in  cm. 
From  His  and  Russow's  data. 


Ill]  OF  GROWTH  IN  INFANCY  113 

porary  set-back  is  immediately  followed  by  a  secondary,  and  equally 
transitory,  acceleration  *. 

Mean  weight  in  grams  of  American  infants  during  ten  days 
after  birth.     (From  Meredith  and  Brown) 

Weight 


Age 

f ^ N 

(days) 

Male 

Femal 

^t  birth 

3491 

3408 

1 

3376 

3283 

2 

3294 

3207 

3 

3274 

3195 

4 

3293 

3213 

5 

3326 

3246 

6 

3366 

3281 

7 

3396 

3315 

8 

3421 

3341 

9 

3440 

3362 

10 

3466 

3387 

Daily 

increment 

A 

Male 

Fema 

-115 

-125 

-  82 

-  76 

-  20 

-  12 

19 

17 

33 

34 

40 

35 

30 

34 

25 

26 

19 

21 

26 

25 

The  set-back  after  birth  of  which  we  have  just  spoken  is  better 
shewn  by  the  child's  weight  than  by  any  linear  measurement.  During 
its  first  three  days  the  infant  loses  weight  visibly,  and  it  is  more  than 
ten  days  old  before  it  has  made  up  the  weight  it  lost  in  those  first 
three  (Fig.  17). 

It  is  worth  our  while  to  illustrate  on  a  larger  scale  His's  careful 
data  for  the  ten  months  of  pre-natal  life  (Fig.  18).  They  give  an 
S-shaped  curve,  beautifully  regular,  and  nearly  symmetrical  on 
either  side  of  its  point  of  inflection;  and  its  differential,  or  curve 
of  monthly  increments,  is  a  bell-shaped  curve  which  indicates  with 
the  utmost  simplicity  a  rise  from  a  minimal  to  a  maximal  rate,  and 
a  fall  to  a  minimum  again.  It  has  a  close  family  Hkeness  to  the 
well-known  "curve  of  probabihty,"  of  which  we  shall  presently 
have  much  more  to  say;  it  is  a  curve  for  which  we  might  well 
hope  to  find  a  simple  mathematical  expression  f- 

These  two  curves,  then,  look  more  "mathematical,"  and  less 
merely  descriptive,  than  any  others  we  have  yet  drawn,  and  much 

*  See  especially,  H.  V.  Meredith  and  A.  W.  Brown,  Growth  in  body-weight 
during  first  ten  days  of  postnatal  life,  Human  Biology,  xi,  pp.  24-77,  1939.  Also 
{int.  al.)  T.  Brailsford  Robertson,  Pre-  and  post-natal  growth,  etc.,  Amer.  Journ. 
Physiol.  XXXVII,  pp.  1^2,  74-85,  1915. 

t  The  same  is  not  less  true  of  Friedenthal's  more  elaborate  measurements,  in  his 
Physiologie  des  Menschenwachstums,  1914;  cf.  Needham,  op.  cit.  p.  1677. 


114 


THE  RATE  OF  GROWTH 

T 


CH. 


4-100 


--100 


Fig.  17. 


0  3  10 

Age  in  days 
Mean  weight  of  American  infants.     From  Meredith  and  Brown's  data. 


500 

^^ 

mm. 

^.-"^"^ 

400 

- 

y  Length 

300 

- 

/ 

1 

200 

J 

1 

100 

- 

leration 

^>^ 

1                      1                      1 

1 

0  2  4  6  8  10 

months 
Fig.  18.     Curve  of  a  child's  pre-natal  growth,  in  length  or  stature;  and  corre- 
sponding curve  of  mean  monthly  increments  (mm.).     (Smoothed  curves.) 


Ill]  OF  GROWTH  IN  INFANCY  115 

the  same  curves  meet  us  again  and  again  in  the  growth  of  other 
organisms.  The  pre-natal  growth  of  the  guinea-pig  is  just  the 
same*.  We  have  the  same  essential  features,  the  same  S-shaped 
curve,  in  the  growth  by  weight  of  an  ear  of  maize  (Fig.  19),  or  the 
growth  in  length  of  the  root  of  a  bean  (Fig.  20);  in  both  we  see 
the  same  slow  beginning,  the  rate  rapidly  increasing  to  a  maximum, 
and  the  subsequent  slowing  down  or  "negative  acceleration "f." 
One  phase  passes  into  another;  so  far  as  these  curves  go,  they 
exhibit  growth  as  a  continuous  process,  with  its  beginning,  its 
middle  and  its  end— a  continuity  which  Sachs  recognised  some 
seventy  years  ago,  and  spoke  of  as  the  "grand  period  of  growth  J." 
But  these  simple  curves  relate  to  simple  instances,  to  the  infant 
sheltered  in  the  womb,  or  to  plant-growth  in  the  sunny  season  of 
the  year.  They  mark  a  favourable  episode,  rather  than  relate  the 
course  of  a  lifetime.  A  curve  of  growth  to  run  all  life  long  is  only 
simple  in  the  simplest  of  organisms,  and  is  usually  a  very  complex 
affair. 

Growth  in  length  of  Vallisneria^,  and  root  ofbean\\ 
and  weight  of  ntaize^ 


VaUisneria 

A 

Vicia 

Zea 

— 

-\ 

Hours           Inches 

Days 

Mm. 

Days 

Gm. 

6                  0-3 

0 

1-0 

6 

1 

16                   1-7 

1 

2-8 

18 

4 

42                 12-6 

2 

6-5 

30 

9 

54                 15-4 

3 

240 

39 

17 

65                 161 

4 

40-5 

46 

26 

77                 16-7 

5 

57-5 

53 

42 

88                 17-1 

6 

72-0 

60 

62 

7 

79-0 

74 

71 

8  790  93  74 

It  would  seem  to  be  a  natural  rule,  that  those  offspring  which 
are  most  highly  organised  at  birth  are  those  which  are  born  largest 

*  See  R.  L.  Draper,  Anat.  Record,  xviii,"p.  369,  1920;  cf.  Needham,  op.  cil., 
p.  1672. 

t  *^f.  R.  Chodat  et  A.  Monnier;  Sur  la  courbe  de  croissance  chez  les  vegetaux. 
Bull.  Herbier  Boissier  (2),  v,  p.  615,  1905. 

X  Arbeiten  a.  d.  bot.  Instit.  Wiirzburg,  i,  p.  569,  1872. 

§  A.  Bennett,  Trans.  Linn.  Soc.  (2),  i  (Bot.),  p.  133,  1880. 

II  Sachs,  I.e. 

^  Stefanowska,  op.  cit. ;  G.  Backman,  Ergebn.  d.  Physiologie,  xxiii,  p.  925,  193j 


116 


THE  RATE  OF  GROWTH 


[CH. 


20  40  60  80  100 

Days 
Fig.  19.     Growth  in  weight  of  maize.     From  Gustav  Backman,  after  Stefanowska. 


0  5  8 

Age  in  days 
Fig.  20.     Growth  in  height  of  a  beanstalk.     From  Sachs's  data. 


Ill]  OF  GROWTH  IN  INFANCY  117 

relatively  to  their  parents'  size.  But  another  rule  comes  in,  which 
is  perhaps  less  to  be  expected,  that  the  offspring  are  born  smaller 
the  larger  the  species  to  which  they  belong.  Here  we  shew,  roughly, 
the  relative  weights  of  the  new-born  animal  and  its  mother*: 


Bear 

1:600 

Sheep 

1:14 

Lion 

160 

Ox 

13 

Hippopotamus 

45 

Horse 

12 

Dog 

45-50 

Rabbit 

40 

Cat 

25 

Mouse  , 

10-25 

Man 

22 

Guinea-pig 

7 

These  differences  at  birth  are  for  the  most  part  made  up  quickly; 
in  other  words,  there  are  great  differences  in  the  rate  of  growth 
during  early  post-natal  life.  Two  Uourcubs,  studied  by  M.  Anthony, 
grew  as  follows: 


Male  Female 


Feb.  23  (born) 

— 

— 

28 

2-0  kilos 

1-7  kilos 

Mar.    8 

30 

2-6 

15 

3-8 

3-3 

22 

4-6 

4-0 

30 

5-3 

4-6 

Apr.    5 

61 

5-2 

12 

7-0 

6-0 

19 

8-0 

7-0 

Thus  the  lion-cub  doubles  its  weight  in  the  first  month,  and 
wellnigh  doubles  it  again  in  the  second;  but  the  newborn  child 
takes  fully  five  months  to  double  its  weight,  and  nearly  two  years 
to  do  so  again. 

The  size  finally  attained  is  a  resultant  of  the  rate  and  of  the 
duration  of  growth;  and  one  or  other  of  these  may  be  the  more 
important,  in  this  case  or  in  that.  It  is  on  the  whole  true,  as  Minot 
said,  that  the  rabbit  is  bigger  than  the  guinea-pig  because  he  grows 
faster,  but  man  is  bigger  than  the  rabbit  because  he  goes  on  growing 
for  a  longer  time. 

A  bantam  and  a  barn-door  fowl  differ  in  their  rate  of  growth, 
which  in  either  case  is  definite  and  specific.  Bantams  have  been 
bred  to  match  almost  every  variety  of  fowl ;  and  large  size  or  small, 
quick  growth  or  slow,  is  inherited  or  transmitted  as  a  Mendehan 

*  Data  from  Variot,  after  Anthony, 


118  OF  THE  RATE  OF  GROWTH  [ch. 

character  in  every  cross  between  a  bantam  and  a  larger  breed. 
The  bantam  is  not  produced  by  selecting  smaller  and  smaller 
specimens  of  a  larger  breed,  as  an  older  school  might  have  supposed ; 
but  always  by  first  crossing  with  bantam  blood,  so  introducing  the 
"character"  of  smallness  or  retarded  growth,  and  then  segregating 
the  desired  types  among  the  dwarfish  offspring.  In  fact.  Darwinian 
selection  plays  a  small  and  unimportant  part  in  the  process*. 

From  the  whole  of  the  foregoing  discussion  we  see  that  rate  of 
growth  is  a  specific  phenomenon,  deep-seated  in  the  nature  of  the 
organism;  wolf  and  dog,  horse  and  ass,  nay  man  and  woman,  grow 
at  different  rates  under  the  same  circumstances,  and  pass  at  different 
epochs  through  like  phases  of  development.  Much  the  same  might 
be  said  of  mental  or  intellectual  growth ;  the  girl's  mind  is  more 
precocious  than  the  boy's,  and  its  development  is  sooner  arrested 
than  the  man's  f. 

On  variability,  and  on  the  curve  of  frequency  or  of  error 

The  magnitudes  which  we  are  dealing  with  in  .this  chapter — 
heights  and  weights  and  rates  of  change — are  (with  few  exceptions) 
mean  values  derived  from  a  large  number  of  individual  cases.  We 
deal  with  what  (to  borrow  a  word  from  atomic  physics)  we  may 
call  an  ensemble;  we  employ  the  equaUsing  powei^  of  averages, 
invoke  the  "law  of  large  numbers  J,"  and  claim  to  obtain  results 
thereby  which  are  more  trustworthy  than  observation  itself  §.  But 
in  ascertaining  a  mean  value  we  must  also  take  account  of  the 
amount  of  variability,  or  departure  from  the  mean,  among  the  cases 
from  which  the  mean  value  is  derived.  This  leads  on  far  beyond 
our  scope,  but  we  must  spare  it  a  passing  word ;  it  was  this  identical 
phenomenon,  in  the  case  of  Man,  which  suggested  to  Quetelet  the 

*  Cf.  Raymond  Pearl,  The  selection  problem,  Amer.  Naturalist,  1917,  p.  82; 
R.  C.  Punnett  and  P.  G.  Bailey,  Journ.  of  Genetics,  iv,  pp.  23-39,  1914. 

t  Cf.  E.  Devaux,  L'p,llure  du  developpement  dans  les  deux  sexes,  Revue  gindr. 
des  Sci.  1926,  p.  598. 

X  S.  D.  Poisson,  following  James  Bernoulli's  Ars  Conjectandi  (op.  posth.  1713), 
was  the  discoverer,  or  inventor,  of  the  law  of  large  numbers.  "Les  chos^s 
de  toute  nature  sont  soumises  a  une  loi  universelle  qu'on  pent  appeler  la  loi  des 
grands  nombres"  {Recherches,  1837,  pp.  7-12). 

§  See  p.  137,  footnote. 


Ill 


OF  VARIABILITY 


119 


statistical  study  of  Variation,  led  Francis  Galton  to  enquire  into 
the  laws  of  Natural  Inheritance,  and  served  Karl  Pearson  as  the 
foundation  of  his  science  of  Biometrics. 

When  Quetelet  tells  us  that  the  mean  stature  of  a  ten-year-old 
boy  is  1-275  metres,  this  is  found  to  imply,  not  only  that  the 
measurements  of  all  his  ten-year-old  boys  group  themselves  about 
this  mean  value  of  1-275  metres,  but  that  they  do  so  in  an  orderly 
way,  many  departing  httle  from  that  mean  value,  and  fewer  and 
fewer  departing  more  and  more.  In  fact,  when  all  the  measure- 
ments are  grouped  and  plotted,  so  as  to  shew  the  number  of 
instances  (y)  ^  each  gradation  of  size  (x),  we  obtain  a  characteristic 


-2cr      —cr 


Fig.  21.     The  normal  curve  of  frequency,  or  of  error. 
a,  -a,  the  "standard  deviation". 

configuration,  mathematically  definable,  called  the  curve  of  frequency, 
or  oi error  (Fig.  21).  This  is  a  very  remarkable  fact.  That  a  "curve 
of  stature"  should  agree  closely  with  the  "normal  curve  of  error" 
amazed  Galton,  and  (as  he  said)  formed  the  mainstay  of  his  long 
and  fruitful  enquiry  into  natural  inheritance*.  The  curve  is  a 
thing  apart,  sui  generis.  It  depicts  no  course  of  events,  it  is  no 
time  or  vector  diagram.  It  merely  deals  with  the  variabihty,  and 
variation,  of  magnitudes;  and  by  magnitudes " we  mean  anything 
which  can  be  counted  or  measured,  a  regiment  of  men,  a  basket  of 


*  Stature  itself,  in  a  homogeneous  population,  is  a  good  instance  of  a  normal 
frequency  distribution,  save  only  that  the  spread  or  range  of  variation  is  unusually 
low;  for  one-half  of  the  population  of  England  differs  by  no  more  than  an  inch 
and  a  half  from  the  average  of  them  all.  Variation  is  said  to  be  greater  among  the 
negroid  than  among  the  white  races,  and  it  is  certainly  very  great  from  one  race 
to  another:  e.g.  from  the  Dinkas  of  the  White  Nile  with  a  mean  height  of  1-8  m. 
to  the  Congo  pygmies  averaging  1-35,  or  say  5  ft.  11  in.  and  4  ft.  6  in.  respectively. 


120  THE  RATE  OF  GROWTH  [ch. 

niits,  the  florets  of  a  daisy,  the  stripes  of  a  zebra,  the- nearness  of 
shots  to  the  bull's  eye*.  It  thereby  illustrates  one  of  the  most 
far-reaching,  some  say  one  of  the  most  fundamental,  of  nature's 
laws. 

We  find  the  curve  of  error  manifesting  itself  in  the  departures 
from  a  mean  value,  which  seems  itself  to  be  merely  accidental — 
as,  for  instance,  the  mean  height  or  weight  of  ten-year-old  English 
boys;  but  we  find  it  no  less  well  displayed  when  a  certain  definite 
or  normal  number  is  indicated  by  the  nature  of  the  case.  For 
instance  the  Medusae,  or  jelly-fishes,  have  a  "radiate  symmetry" 
of  eight  nodes  and  internodes.  But  even  so,  the  number  eight  is 
subject  to  variation,  and  the  instances  of  more  or  less  graup  them- 
selves in  a  Gaussian  curve. 

Number  of  "  tentaculocysts''  in  Medusae  {Ephyra  and  Aurelia) 
[Data  from  E.  T.  Browne,  QJ.M.S.  xxxvii,  p.  245,  1895) 


5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

Ephyra  (1893)      — 
„       (1894)        1 

4 
6 

8 
34 

278 
883 

22 

75 

18 
61 

12 
35 

14 
17 

3 
3 

1 

— 

Aurelia  (1894)      — 

2 

18 

296 

33 

16 

18 

7 

— 

— 

1 

Percentage  numbers: 
Ephyra                  — 

M 

0-5 

2-2 
30 

77-4 
79-0 

61 

6-7 

5-0 
5-4 

3-3 
31 

3-9 
1-4 

0-8 
0-2 

— 

— 

Aurelia                 — 

0-5 

4-7 

77-2 

8-6 

41 

2-6 

1-8 

— 

— 

— 

Mean          — 

0-7 

3-3 

77-9 

7-1 

4-8 

3-0 

2-4 

0-3 

— 

— 

The  curve  of  error  is  a  "bell-shaped  curve,"  a  courbe  en  cloche.  It 
rises  to  a  maximum,  falls  away  on  either  side,  has  neither  beginning 
nor  end.  It  is  (normally)  symmetrical,  for  lack  of  cause  to  make  it 
otherwise;  it  falls  off  faster  and  then  slower  the  farther  it  departs 
from  the  mean  or  middle  line;  it  has  a  "point  of  inflexion,"  of 
necessity  on  either  side,  where  it  changes  its  curvature  and  from 
being  concave  to  the  middle  line  spreads  out  to  become  convex 

*  "I  know  of  scarcely  anything  (says  Galton)  so  apt  to  impress  the  imagination 
as  the  wonderful  form  of  cosmic  order  expressed  by  the  Law  of  Frequency  of 
Error. ...  It  reigns  with  serenity  and  in  complete  self-effacement  amidst  the 
wildest  confusion"  {Natural  Inheritance,  p.  62).  Observe  that  Galton  calls  it  the 
"law  of  frequency  o/  error,"  which  is  indeed  its  older  and  proper  name.  Cf.  (int.  al.) 
P.  G.  Tait,  Trans  R.S.E.  xxiv,  pp.  139-145,  1867. 


i 


Ill]  OF  THE  CURVE  OF  ERROR  121 

thereto.  If  we  pour  a  bushel  of  corn  out  of  a  sack,  the  outhne  or 
profile  of  the  heap  resembles  such  a  curve ;  and  wellnigh  every  hill 
and  mountain  in  the  world  is  analogous  (even  though  remotely)  to 
that  heap  of  corn  *.  Causes  beyond  our  ken  have  cooperated  to  place 
and  allocate  each  grain  or  pebble;  and  we  call  the  result  a  "random 
distribution,"  and  attribute  it  to  fortuity,  or  chance.  Galton 
devised  a  very  beautiful  experiment,  in  which  a  slopmg  tray  is 
beset  with  pins,  and  sand  or  millet-seed  poured  in  at  the  top. 
Every  falhng  grain  has  its  course  deflected  again  and  again;  the 
final  distribution  is  emphatically  a  random  one,  and  the  curve  of 
error  builds  itself  up  before  our  eyes. 

The  curve  as  defined  by  Gauss,  princeps  mathematicorum — who 
in  turn  was  building  on  Laplace  f — is  at  once  empirical  and 
theoretical  J ;  and  Lippmann  is  said  to  have  remarked  to  Poincare : 
*'Les  experimentateurs  s'imaginent  que  c'est  un  theoreme  de 
mathematique,  et  les  mathematiciens  d'etre  un  fait  experimental ! " 
It  is  theoretical  in  so  far  as  its  equation  is  based  on  certam  hypo- 
thetical considerations:  viz.  (1)  that  the  arithmetic  mean  of  a  number 
of  variants  is  their  best  or  likeliest  average,  an  axiom  which  is 
obviously  true  in  simple  cases — but  not  necessarily  in  all;  (2)  that 
"fortuity"  implies  the  absence  of  any  predominant,  decisive  or 
overwhelming  cause,  and  connotes  rather  the  coexistence  and  joint 
effect  of  small,  undefined  but  independent  causes,  many  or  few: 

*  If  we  pour  the  corn  out  carefully  through  a  small  hole  above,  the  heap  becomes 
a  cone,  with  sides  sloping  at  an  "angle  of  repose";  and  the  cone  of  Fujiyama  is  an 
exquisite  illustration  of  the  same  thing.  But  in  these  two  instances  one  predominant 
cause  outweighs  all  the  rest,  and  the  distribution  is  no  longer  a  random  one. 

t  The  Gaussian  curve  of  error  is  really  the  "second  curve  of  error"  of  Laplace. 
Laplace's  first  curve  of  error  (which  has  uses  of  its  own)  consists  of  two  exponential 
curves,  joining  in  a  sharp  peak  at  the  median  value.  Cf.  W.  J.  Luyten,  Proc. 
Nat.  Acad.  Sci.  xvm,  pp.  360-365,  1932. 

I  The  Gaussian  equation  to  the  normal  frequency  distribution  or  "curve  of 
error"  need  not  concern  us  further,  but  let  us  state  it  once  for  all: 

J,        _(xa-x)* 

^        V27T 

where  Xf^  is  the  abscissa  which  gives  the  maximum  ordinate,  and  where  the  maximum 
ordinate,  y^  =  1/^/(27t).  Thus  the  log  of  the  ordinate  is  a  quadratic  function  of  the 
abscissa ;  and  a  simple  property,  fundamental  to  the  curve,  is  that  for  equally  spaced 
ordinates  (starting  anywhere)  the  square  of  any  ordinate  divided  by  the  product  of 
its  neighbours  gives  a  scalar  quantity  which  is  constant  all  along  (G.T.B.). 


122  THE  RATE  OF  GROWTH  [ch. 

producing  their  several  variations,  deviations  or  errors;  and  potent 
in  their  combinations,  permutations  and  interferences*. 

We  begin  to  see  why  bodily  dimensions  lend,  or  submit,  them- 
selves to  this  masterful  law.  Stature  is  no  single,  simple  thing; 
it  is  compounded  of  bones,  cartilages  and  other  elements,  variable 
each  in  its  own  way,  some  lengthening  as  others  shorten,  each 
playing  its  little  part,  hke  a  single  pin  in  Galton's  toy,  towards  a 
''fortuitous"  resultant.  "The  beautiful  regularity  in  the  statures 
of  a  population  (says  Galton)  whenever  they  are  statistically 
marshalled  in  the  order  of  their  heights,  is  due  to  the  number  of 
variable  and  quasi-independent  elements  of  which  stature  is  the 
sum."  In  a  bagful  of  pennies  fresh  from  the  Mint  each  coin  is 
made  by  the  single  stroke  of  an  identical  die,  and  no  ordinary 
weights  and  measures  suffice  to  differentiate  them;  but  in  a  bagful 
of  old-fashioned  hand-made  nails  a  slow  succession  of  repeated 
operations  has  drawn  the  rod  and  cut  the  lengths  and  hammered 
out  head,  shaft  and  point  of  every  single  nail — and  a  curve  of 
error  depicts  the  differences  between  them. 

The  law  of  error  was  formulated  by  Gauss  for  the  sake  of  the 
astronomers,  who  aimed  at  the  highest  possible  accuracy,  and 
strove  so  to  interpret  their  observations  as  to  eliminate  or  minimise 
their  inevitable  personal  and  instrumental  errors.  It  had  its 
roots  also  in  the  luck  of  the  gaming-table,  and  in  the  discovery 
by  eighteenth-century  mathematicians  that  "chance  might  be 
defined  in  terms  of  mathematical  precision,  or  mathematical  'law'." 
It  was  Quetelet  who,  beginning  as  astronomer  and  meteorologist, 
applied  the  "law  of  frequency  of  error"  for  the  first  time  to 
biological  statistics,  with  which  in  name  and  origin  it  had  nothing 
whatsoever  to  do. 

The  intrinsic  significance  of  the  theory  of  probabihties  and  the 
law  of  error  is  hard  to  understand.  It  is  sometimes  said  that  to 
forecast  the  future  is  the  main  purpose  of  statistical  study,  and 
expectation,  or  expectancy,  is  a  common  theme.     But  all  the  theory 

*  "The  curve  of  error  would  seem  to  carry  the  great  lesson  that  the  ultimate 
differences  between  individuals  are*  simple  and  few ;  that  they  depend  on  collisions 
and  arrangements,  on  permutations  and  combinations,  on  groupings  and  inter- 
ferences, of  elementary  qualities  which  are  limited  iyi  variety  and  finite  in  extent'' 
(J.  M.  Keynes).  A  connection  between  this  law  and  Mendelian  inheritance  is 
discussed  by  John  Brownlee,  P.R.S.E.  xxxi,  p.  251,  1910. 


Ill]  bF  THE  LAW  OF  ERROR  123 

in  the  vorld  enables  us  to  foretell  no  single  unknown  thing,  not  even 
the  turn  of  a  card  or  the  fall  of  a  die.  The  theory  of  probabiUties 
is  a  development  of  the  theory  of  combinations,  and  only  deals  with 
what  occurs,  or  has  occurred,  in  the  long  run,  among  large  numbers 
and  many  permutations  thereof.  Large  numbers  simplify  many 
things;  a  million  men  are  easier  to  understand  than  one  man  out 
of  a  million.  As  David  Hume*  said:  "What  depends  on  a  few 
persons  is  in  a  great  measure  to  be  ascribed  to  chance,  or  to  secret 
and  unknown  causes;  what  arises  from  a  great  many  may  often 
be  accounted  for  by  determinate  and  known  causes."  Physics  is, 
or  has  become,  a  comparatively  simple  science,  just  because  its  laws 
are  based  on  the  statistical  averages  of  innumerable  molecular  or 
primordial  elements.  In  that  invisible  world  we  are  sometimes  told 
that  "chance"  reigns,  and  "uncertainty"  is  the  rule;  but  such 
phrases  as  mere  chance,  or  at  random,  have  no  meaning  at  all  except 
with  reference  to  the  knowledge  of  the  observer,  and  a  thing  is  a 
"pure  matter  of  chance"  when  it  depends  on  laws  which  we  do  not 
know,  or  are  not  considering  f.  Ever  since  its  inception  the  merits 
and  significance  of  the  theory  of  probabiUties  have  been  variously 
estimated.  Some  say  it  touches  the  very  foundations  of  know- 
ledge} ;  and  others  remind  us  that  "avec  les  chiffres  on  pent  tout 
demontrer."  It  is  beyond  doubt,  it  is  a  matter  of  common  ex- 
perience, that  probability  plays  its  part  as  a  guide  to  reasoning. 
It  extends,  so  to  speak,  the  theory  of  the  syllogism,  and  has  been 
called  the  "logic  of  uncertain  inference "§. 

In  measuring  a  group  of  natural  objects,  our  measurements  are 
uncertain  on  the  one  hand  and  the  objects  variable  on  the  other; 
and  our  first  care  is  to  measure  in  such  a  way,  and  to  such  a  scale, 
that  our  own  errors  are  small  compared  with  the  natural  variations. 
Then,  having  made  our  careful  measurements  of.  a  group,  we  want 
to  know  more  of  the  distribution  of  the  several  magnitudes,  and 


*  Essay  xiv. 

t  So  Leslie  Ellis  and  G.  B.  Airy,  in  correspondence  with  Sir  J.  D.  Forbes;  see 
his  Life,  p.  480. 

X  Cf.  Hans  Reichenbach,  Les  fondements  logiques  du  calcul  des  probabilit^s, 
Annales  de  Vinst.  Poincare,  vii,  pp.  267,  1937. 

§  Cf.  J.  M.  Keynes,  A  Treatise  on  Probability,  1921;  and  A.  C.  Aitken's  Statistical 
Mathematics,  1939. 


124  THE  RATE  OF  GROWTH  [ch. 

especially  to  know  two  important  things.  We  want  a  mean  value, 
as  a  substitute  for  the  true  value*  if  there  be  such  a  thing;  let  us 
use  the  arithmetic  mean  to  begin  with.  About  this  mean  the  ob- 
served values  are  grouped  like  a  target  hit  by  skilful  or  unskilful 
shots ;  we  want  some  measure  of  their  inaccuracy,  some  measure  of 
their  spread,  or  scatter,  or  dispersion,  and  there  are  more  ways  than 
one  of  measuring  and  of  representing  this.  We  do  it  visibly  and 
graphically  every  time  we  draw  the  curve  (or  polygon)  of  frequency ; 
but  we  want  a  means  of  description  or  tabulation,  in  words  or  in 
numbers.  We  find  it,  according  to  statistical  mathematics,  in  the 
so-called  index  of  variability,  or  standard  deviation  (o),  which  merely 
means  the  average  deviation  from  the  meanf.  But  we  must  take 
some  precautions  in  determining  this  average;  for  in  the  nature  of 
things  these  deviations  err  both  by  excess  and  defect,  they  are 
partly  positive  and  partly  negative,  and  their  mean  value  is  the  mean 
of  the  variants  themselves.  Their  squares,  however,  are  all  positive, 
and  the  mean  of  these  takes  account  of  the  magnitude  of  each 
deviation  with  no  risk  of  cancelling  out  the  positive  and  negative 
terms:  but  the  "dimension  "  of  this  average  of  the  squares  is  wrong. 
The  square  root  of  this  average  of  squares  restores  the  correct 
dimension,  and  the  result  is  the  useful  index  of  variability,  or  of 
deviation,  which  is  called  o-J. 

This  standard  deviation  divides  the  area  under  the  normal  curve 
nearly  into  equal  halves,  and  nearly  coincides  with  the  point  of 
inflexion  on  either  side;  it  is  the  simplest  algebraic  measure  of 
dispersion,  as  the  mean  is  the  simplest  arithmetical  measure  of 
position.     When  we  divide  this  value  by  the  mean,  we  get  a  figure 


*  It  is  not  always  obvious  what  the  "errors"  are,  nor  what  it  is  that  they  depart 
or  deviate  from.  We  are  apt  to  think  of  the  arithmetic  mean,  and  to  leave  it 
at  that.  But  were  we  to  try  to  ascertain  the  ratio  of  circumference  to  diameter 
by  measuring  pennies  or  cartwheels,  our  "errors"  would  be  found  grouped  round 
a  mean  value  which  no  simple  arithmetic  could  define. 

t  a,  the  standard  deviation,  was  chosen  for  its  convenience  in  mathematical 
calculation  and  formulation.  It  has  no  special  biological  significance;  and  a 
simpler  index,  the  "inter-quartile  distance,"  has  its  advantages  for  the  non- 
mathematician,  as  we  shall  see  presently. 

X  That  is  to  say :  Square  the  deviation-from-the-mean  of  each  class  or  ordinate 
(^);    multiply  each  by  the  number  of  instances  (or  "variates")  in  ^that  class  (/); 

divide  by  the  total  number  (N) ;  and  take  the  square-root  of  the  whole :  a^=  ~  "^  *'    . 


I 


Ill]  THE  COEFFICIENT  OF  VARIABILITY  125 

which  is  independent  of  any  particular  units,  and  which  is  called 
the  coefficient  of  variability''^ . 

Karl  Pearson,  measuring  the  amount  of  variability  in  the  weight 
and  height  of  man,  found  this  coefficient  to  run  as  follows :  In  male 
new-born  infants,  for  weight  15-6,  and  for  stature  6-5;  in  male 
adults,  for  weight  10-8,  and  for  stature  3-6.  Here  the  amount  of 
variability  is  thrice  as  great  for  weight  as  for  stature  among  grown 
men,  and  about  2 J  times  as  great  in  infancy  f.  The  same  curious 
fact  is  well  brought  out  in  some  careful  measurements  of  shell-fish, 
as  follows; 

Variability  of  youdg  Clams  (Mactra  sp.)X 


Average  size 

A 

Coefficient  of 
variability 

Age  (years) 
Number  in  sample 

1                   2 
41                 20 

1                    2 
41                  20 

Length  (cm.) 

Height 

Thickness 

3-2                6-3 
2-3                4-7 
1-3                2-8 

15-3                6-3 
140                6-7 

9-6                8-3 

Weight  (gm.) 

6-4              59-8 

35-4               18-5 

The  phenomenon  is  purely  mathematical.  Weight  varies  as  the  product  of 
length,  height  and  depth,  or  (as  we  have  so  often  seen)  as  the  cube  of  any  one 
of  these  dimensions  in  the  case  of  similar  figures.  It  is  then  a  mathematical, 
rather  than  a  biological  fact  that,  for  small  deviations,  the  variability  of  the 
whole  tends  to  be  equal  to  the  sum  of  that  of  the  three  constituent  dimensions. 
For  if  weight,  w,  varies  as  height  x,  breadth  y,  and  depth  z,  we  may  write 

w  =  c.xyz. 

„„  ,.„        ,.  ,.  dw     dx     dy     dz 

Whence,  ditterentiatmg,  = h  -^  H . 

w       X       y       z 

We  see  that  among  the  shell-fish  there  is  much  more  variability 
in  the  younger  than  in  the  older  brood.     This  may  be  due  to 

*  It  is  usually  multiplied  by  100,  to  make  it  of  a  handier  amount;  and  we  may 
then  define  this  coefficient,  C,  AS  —  ajM  x  100. 

t  Cf.  Fr.  Boas,  Growth  of  Toronto  children,-  Rep.  of  U.S.  Cornm.  of  Education, 
1896-7,  1898,  pp.  1541-1599;  Boas  and  Clark  Wissler,  Statist  cs  of  growth, 
Education  Rep.  1904,  1906,  pp.  25-132;  H.  P.  Bowditch  Rep.  Mass.  State  Board 
of  Health,  1877;  K.  Pearson,  On  the  magnitude  of  certain  coefficients  of  correlation 
in  man,  Proc.  R.S.  lxvi,  1900;  S.  Nagai,  Korperkonstitution  der  Japaner,  from 
Brugsch-Levy,  Biologie  d.  Person,  ii,  p.  445,  1928;  R.  M.  Fleming,  A  study  of 
growth  and  development.  Medical  Research  Council,  Special  Report,  No.  190,  1933. 

J  From  F.  W.  Weymouth,  California  Fish  Bulletin,  No.  7,  1923. 


126  THE  RATE  OF  GROWTH  [ch. 

inequality  of  age;  for  in  a  population  only  a  few  weeks  old,  a  few 
days  sooner  or  later  in  the  date  of  birth  would  make  more  diiference 
than  later  on.  But  a  more  important  matter,  to  be  seen  in  man- 
kind (Fig.  22),  is  that  variability  of  stature  runs  j)ari  passu,  or 
nearly  so,  with  the  rate  of  growth,  or  curve  of  annual  increments 
(cf.  Fig.  12).  The  curve  of  variability  descends  when  the  growth- 
rate  slackens,  and  rises  high  when  in  late  boyhood  growth  is  speeded 
up.  In  short,  the  amount  of  variability  in  stature  or  in  weight  is 
correlated  with,  or  is  a  function  of,  the  rate  of  growth  in  these 
magnitudes. 

Judging  from  the  evidence  at  hand,  we  may  say  that  variabihty 
reaches  its  height  in  man  about  the  age  of  thirteen  or  fourteen, 
rather  earlier  in  the  girls  than  in  the  boys,  and  rather  earher  in  the 
case  of  stature  than  of  weight.  The  difference  in  this  respect  between 
the  boys  and  the  girls  is  now  on  one  side,  now  on  the  other.  In 
infancy  variabihty  is  greater  in  the  girls;  the  boys  shew  it  the 
more  at  five  or  six  years  old;  about  ten  years  old  the  girls  have 
it  again.  From  twelve  to  sixteen  the  boys  are  much  the  more 
variable,  but  by  seventeen  the  balance  has  swung  the  other  way 
(Fig.  23). 

Coefficient  of  variability  (ojM  x  100)  in  man,  at  various  ages 
Age  ...     o        6        7        8        9       10       11       12       13       14       15       16       17      18 

ytature 


I 


British  (Fleming): 

Boys 

•51 

5-4 

50 

5-3 

5-4 

5-6 

5-7 

5-6 

5-8 

5-8 

5-8 

5-0 

4-3 

30 

Girls 

5-2/ 

5-2 

50 

5-5 

5-4 

5-6 

5-8 

5-7 

5-6 

4-7 

4-2 

3-9 

3-7 

3-8 

American 

4-8 

4-6 

4-4 

4-5 

4-4 

4-6 

4-7 

4-9 

5-5 

5-8 

5-6 

5-5 

4-6 

3-7 

(Bowditch) 

Japanese  (Nagai): 

Boys 



40 

— 

4-3 

— 

41 

— 

40 

50 

50 

4-2 

3-2 

— 

— 

Girls 

— 

4-3 

— 

41 

— 

4-5 

— 

4-5 

4-6 

3-6 

31 

30 

— 

— 

Mean 

— 

4-7 

— 

4-7 

— 

4-9 

— 

^  5-0 

5-3 

50 

4-6 

41 

— ■ 

— 

Weight 

American 

11-6 

10-3 

Ill 

9-9 

110 

1-6 

1-8 

13-7 

3-6 

6-8 

15-3 

13-3 

130 

10-4 

Japanese : 

Boys 



10-3 



121 

— 

0-8 

— 

7-0 

51 

70 

13-8 

10-9 

— 

— 

Girls 

— 

10-2 

— 

11-2 

— 

21 

— 

150 

5-6 

3-4 

11-4 

11-5 

— 

— 

Mean        —    10-3      —    111      —    11-5      —    11-9   14-8   15-7    13-5    11-9      —      -- 


Ill]  THE  COEFFICIENT  OF  VARIABILITY  127 


61 1 1 1 1 1 r 


1  ^ 


y  r^ 


American.-^         / 
-^  / 


Japanese^ 


I         I         I         I         I \ L 


5      6      7      8      9     10     II     12     13     14     15     16     17     II 
Age 
Fig.  22.     Variability  in  stature  (boys).     After  Fleming,  Bowditch  and  Nagai. 


+  z 

[— 

+  1 

t 

- 

B 

ntish 

/  / 
/  / 

// 

// 

^     \ 
\     \ 

\     \ 
\     \ 
\     \ 
\     \ 

/ 

/ 

/ 
/ 

\ 

-1 

/ 

1 

Japanese 

1 

1 

3 


18 


.  10  15 

Age  in  years 
Fig.  23.     Coefficient  of  variability  in  stature:  excess  or  defect  of  this  coefficient 
in  the  boy  over  the  girl.     Data  from  R.  M.  Fleming,  and  from  Nagai. 


128  THE  RATE  OF  GROWTH  [ch. 

The  amount  of  variability  is  bound  to  differ  from  one  race  or 
nationality  to  another,  and  we  find  big  differences  between  the 
Americans  and  the  Japanese,  both  in  magnitude  and  phase  (Fig.  22). 

If  we  take  not  merely  the  variability  of  stature  or  -weight  at  a 
given  age,  but  the  variability  of  the  yearly  increments,  we  find 
that  this  latter  variabihty  tends  to  increase  steadily,  and  more  and 
more  rapidly,  within  the  ages  for  which  we  have  information;  and 
this  phenomenon  is,  in  the  main,  easy  of  explanation.  For  a  great 
part  of  the  difference  between  one  individual  and  another  in  regard 
to  rate  of  growth  is  a  mere  difference  of  phase — a  difference  in  the 
epochs  of  acceleration  and  retardation,  and  finally  a  difference  as  to 
the  epoch  when  growth  comes  to  an  end ;  it  follows  that  variabihty 
will  be  more  and  more  marked  as  we  approach  and  reach  the  period 
when  some  individuals  still  continue,  and  others  have  already  ceased, 
to  grow.  In  the  following  epitomised  table,  I  have  taken  Boas's 
determinations  *  of  the  standard  deviation  (ct),  converted  them  into 
the  corresponding  coefficients  of  variabihty  {olM  x  100),  and  then 
smoothed  the  resulting  numbers: 

Coefficients  of  variability  in  annual  increments  of  stature 


Age    ... 

7 

8 

9 

10 

11 

12 

13 

14 

15 

Boys 
Girls 

17-3 
171 

15-8 
17-8 

18-6 
19-2 

191 
22-7 

21-0 
25-9 

24-7 
29-3 

29-0 
370 

36-2 
44-8 

461 

The  greater  variabihty  in  the  girls  is  very  marked  f,  and  is 
explained  (in  part  at  least)  by  the  jnore  rapid  rate  at  which  the  girls 
run  through  the  several  phases  of  their  growth  (Fig.  24).  To  say  that 
children  of  a  given  age  vary  in  the  rate  at  which  they  are  growing 
would  seem  to  be  a  more  fundamental  statement  than  that  they 
vary  in  the  size  to  which  they  have  grown. 

Just  as  there  is  a  marked  difference  in  phase  between  the  growth- 
curves  of  the  two  sexes,  that  is  to  say  a  difference  in  the  epochs 
when  growth  is  rapid  or  the  reverse,  so  also,  within  each  sex,  will 
there  be  room  for  similar,  but  individual,  phase-differences.  Thus 
we  may  have  children  of  accelerated  development,  who  at  a  given 

*  Op.  cit.  p.  lo48. 

I  That  women  are  on  the  whole  more  variable  than  men  was  argued  by  Karl 
Pearson  in  one  of  his  earlier  essays:   The  Chances  of  Death  and  other  Studies,  1897. 


in 


THE  COEFFICIENT  OF  VARIABILITY 


129 


epoch  after  birth  are  growing  rapidly  and  are  already  "big  for  their 
age";  and  .others,  of  retarded  development,  who  are  comparatively 
small  and  have  not  reached  the  period  of  acceleration  which,  in 
greater  or  less  degree,  will  come  to  them  in  turn.  In  other  words, 
there  must  under  such  circumstances  be  a  strong  positive  "coefl&cient 
of  correlation"  between  stature  and  rate  of  growth,  and  also  between 


§    20 


7         8         9        10        II        12        13        14       15 

Age  in  years 

Fig.  24.     Coefficients  of  variability,  in  annual  increments  of  stature. 
After  Boas. 

the  rate  of  growth  in  one  year  and  the  next.  But  it  does  not  by 
any  means  follow  that  a  child  who  is  precociously  big  will  continue 
to  grow  rapidly,  and  become  a  man  or  woman  of  exceptional 
stature  *.  On  the  contrary,  when  in  the  case  of  the  precocious  or 
"accelerated"  children  growth  has  begun  to  slow  down,  the  back- 


*  Some  first  attempts  at  analysis  seem  to  shew  that  the  size  of  the  embryo  at 
birth,  or  of  the  seed  at  germination,  has  more,  influence  than  we  were  wont  to 
suppose  on  the  ultimate  size  of  plant  or  animal.  See  (e.g.)  Eric  Ashby,  Heterosis 
and  the  inheritance  of  acquired  characters,  Proc.  R.S.  (B),  No.  833,  pp.  431-441, 
1937;  and  papers  quoted  therein. 


130  THE  RATE  OF  GROWTH  [ch. 

ward  ones  may  still  be  growing  rapidly,  and  so  making  up  (more 
or  less  completely)  on  the  others.  In  other  words,  the  period  of 
high  positive  correlation  between  stature  and  increment  will  tend 
to  be  followed  by  one  of  negative  correlation.  This  interesting  and 
important  point,  due  to  Boas  and  Wissler*,  is  confirmed  by  the 
following  table : 

Correlation  of  stature  and  increment  in  boys  and  girls 
(From  Boas  and  Wissler) 

Age     6  7  8  9  10  11  12  13  14  15 

Stature        (B)  112-7  115-5  123-2     127-4  133-2  136-8  142-7     147-3     155-9  162-2 

(G)  111-4  117-7  121-4     127-9  131-8  136-7  144-6     149-7     153-8  157-2 

Increment  (B)  5-7  5-3  4-9        5-1  5-0  4-7  5-9        7-5        6-2  5-2 

(G)  5-9  5-5  5-5         5-9  6-2  7-2  6-5         5-4        3-3  1-7 

Correlation  (B)  0-25  0-11  0-08      0-25  018  0-18  0-48      0-29    -0-42  -0-44 

(G)  0-44  0-14  0-24      0-47  018  -0-18  -0-42    -0-39    -0-63  0-11 

A  minor  but  very  curious  point  brought  out  by  the  same 
investigators  is  that,  if  instead  of  stature  we  deal  with  height  in 
the  sitting  posture  (or,  practically  speaking,  with  length  of  trunk 
or  back),  then  the  correlations  between  this  height  and  its  annual 
increment  are  throughout  negative.  In  other  words,  there  would 
seem  to  be  a  general  tendency  for  the  long  trunks  to  grow  slowly 
throughout  the  whole  period  under  investigation.  It  is  a  well- 
known  anatomical  fact  that  tallness  is  in  the  main  due  not  to  length 
of  body  but  to  length  of  limb. 

Since  growth  in  height  and  growth  in  weight  have  each,  their  own 
velocities,  and  these  fluctuate,  and  even  the  amount  of  their 
variabihty  alters  with  age,  it  follows  that  the  correlation  between 
height  and  weight  must  not  only  also  vary  but  must  tend  to 
fluctuate  in  a  somewhat  complicated  way.  The  fact  is,  this  corre- 
lation passes  through  alternate  maxima  and  minima,  chief  among 
which  are  a  maximum  at  about  fourteen  years  of  age  and  a  minimum 
about  twenty-one.  Other  intercorrelations,  such  as  those  between 
height  or  weight  and  chest-measurement,  shew  their  periodic 
variations  in  like  manner;    and  it  is  about  the  time  of  puberty 

*  I.e.  p.  42,  and  other  papers  there  quoted.  Cf.  also  T.  B.  Robertson,  Criteria 
of  Normality  in  the  Growth  of  Children,  Sydney,  1922. 


Ill]  THE  CURVE  OF  ERROR  131 

that  correlation  tends  to  be  closest,  or  a  norm  to  be  most  nearly 
approached*. 

The  whole  subject  of  variabiHty,  both  of  magnitude  and  rate  of 
increment,  is  highly  suggestive  and  instructive:  inasmuch  as  it 
helps  further  to  impress  upon  us  that  growth  and  specific  rate  of 
growth  are  the  main  physiological  factors,  of  which  specific  mag- 
nitude, dimensions  and  form  are  the  concrete  and  visible  resultant. 
Nor  may  we  forget  for  a  moment  that  growth-rate,  and  growth 
itself,  are  both  of  them  very  complex  things.  The  increase  of  the 
active  tissues,  the  building  of  the  skeleton  and  the  laying  up 
of  fat  and  other  stores,  all  these  and  more  enter  into  the  complex 
phenomenon  of  growth.  In  the  first  instance  we  may  treat  these 
many  factors  as  though  they  were  all  one.  But  the  breeder  and 
the  geneticist  will  soon  want  to  deal  with  them  apart;  and  the 
mathematician  will  scarce  look  for  a  simple  expression  where 
so  many  factors  are  involved.  But  the  problems  of  variability, 
though  they  are  intimately  related  to  the  general  problem  of 
growth,  carry  us  very  soon  beyond  our  hmitations. 

The  curve  of  error 

To  return  to  the  curve  of  error. 

The  normal  curve  is  a  symmetrical  one.  Its  middle  point,  or 
median  ordinate,  marks  the  arithmetic  mean  of  all  the  measurements ; 
it  is  also  the  mode,  or  class  to  which  the  largest  number  of  individual 
instances  belong.  Mean,  median  and  mode  are  three  diiferent  sorts 
of  average;  but  they  are  one  and  the  same  in  the  normal  curve. 

It  is  easy  to  produce  a  related  curve  which  is  not  symmetrical, 
and  in  which  mean,  median  and  mode  are  no  longer  the  same. 
The  heap  of  corn  will  be  lop-sided  or  "skew"  if  the  wind  be  blowing 
while  the  grain  is  falling:  in  other  words,  if  some  prevailing  cause 
disturb  the  quasi-equihbrium  of  fortuity ;  and  there  are  other  ways, 
some  simple,  some  more  subtle,  by^  which  asymmetry  may  be 
impressed  upon  our  curve. 

The  Gaussian  curve  is  only  one  of  many  similar  bell-shaped  curves ; 
and  the  binomial  coefficients,  the  numerical  coefficients  of  (a  +  by, 
yield  a  curve  so  Hke  it  that  we  may  treat  them  as  the  same.     The 

*  Cf.  Joseph  Bergson,  Growth -changes  in  physical  correlation,  Human  Biology, 
I,  p.  4,  1930. 


132  THE  RATE  OF  GROWTH  [ch. 

Gaussian  curve  extends,  in  theory,  to  infinity  at  either  end;  and 
this  infinite  extension,  or  asymptotism,  has  its  biological  significance. 
We  know  that  this  or  that  athletic  record  is  lowered,  slowly  but 
continually,  as  the  years  go  by.  This  is  due  in  part,  doubtless,  to 
increasing  skill  and  improved  technique ;  but  quite  apart  from  these 
the  record  would  slowly  fall  as  more  and  more  races  are  run,  owing 
to  the  indefinite  extension  of  the  Gaussian  curve*. 

On  the  other  hand,  while  the  Gaussian  curve  extends  in  theory 
to  infinity,  the  fact  that  variation  is  always  limited  and  that  extreme 
v3,riations  are  infinitely  rare  is  one  of  the  chief  lessons  of  the  law 
of  frequency.  If,  in  a  population  of  100,000  men,  170  cm.  be  the 
mean  height  and  6  cm.  the  standard  deviation,  only  11  per  cenl^., 
or  say  130  men,  will  exceed  188  cm.,  only  10  men  will  be  over 
191  cm.,  and  only  one  over  193  cm.,  or  13 J  per  cent,  above  the 
average.  The  chance  is  negligible  of  a  single  one  being  found  over 
210  cm.,  or  7  ft.  high,  or  24  per  cent,  above  the  average. 

Yet,  widely  as  the  law  holds  good,  it  is  hardly  safe  to  count  it 
as  a  universal  law.  Old  Parr  at  150  years  old,  or  the  giant  Chang 
at  more  than  eight  feet  high,  are  not  so  much  extreme  instances  of 
a  law  of  probabihty,  as  exceptional  cases  due  to  some  peculiar  cause 
or  influence  coming  inf.  In  a  somewhat  analogous  way,  one  or  two 
species  in  a  group  grow  far  beyond  the  average  size ;  the  Atlas  moth, 
the  Gohath  beetle,  the  ostrich  and  the  elephant,  are  far-off  outhers 
from  the  groups  to  which  they  belong.  A  reason  is  not  easy  to. find. 
It  looks  as  though  variations  came  at  last  to  be  in  proportion  to  the 
size  attained,  and  so  to  go  on  by  compound  interest  or  geometrical 
progression.  There  may  be  nothing  surprising  in  this ;  nevertheless, 
it  is  in  contradistinction  to  that  summation  of  small  fortuitous 
differences  which  lies  at  the  root  of  the  law  of  error.  If  size  vary  in 
proportion  to  the  magnitude  of  the  variant  individuals,  not  only 


*  This  is  true  up  to  a  certain  extent,  but  would  become  a  mathematical  fiction 
later  on.  There  will  be  physical  limitations  (as  there  are  in  quantum  mechanics) 
both  to  record-breaking,  and  to  the  measurement  of  minute  extensions  of  the 
record. 

t  We  may  indeed  treat  old  Parr's  case  on  the  ordinary  lines  of  actuarial 
probability,  but  it  is  "without  much  actuarial  importance."  The  chance  of  his 
record  being  broken  by  a  modern  centenarian  is  reckoned  at  (5)^°,  by  Major 
Greenwood  and  J.  C.  Irwin,  writing  on  Senility,  in  Human  Biology,  xi,  pp.  1-23, 
1939. 


Ill] 


THE  CURVE  OF  ERROR 


133 


will  the  frequency  curve  be  obviously  skew,  but  the  geometric  mean^ 
not  the  arithmetic,  becomes  the  most  probable  value*.  Now  the 
logarithm  of  the  geometric  mean  of  a  series  of  numbers  is  the 
arithmetic  mean  of  their  logarithms;  and  it  follows  that  in  such 
cases  the  logarithms  of  the  variants,  and  not  the  variants  them- 
selves, will  tend  to  obey  the  Gaussian  law  and  follow  the  normal 
curve  of  frequency  f. 

The  Gaussian  curve,  and  the  standard  deviation  associated  with 
it,  were  (as  we  have  seen)  invented  by  a  mathematician  for  the  use 


21 


33 


36 


24  27  30 

Length  in  mm. 
Fig.  25  A.     Curve  of  frequency  of  a  population  of  minnows. 


39 


of  an  astronomer,  and  their  use  in  biology  has  its  difficulties  and 
disadvantages.  We  may  do  much  in  a  simpler  way.  Choosing  a 
random  example,  I  take  a  catch  of  minnows,  measured  in  3  mm. 
groups,  as  follows  (Fig.  25 A): 

Size  (mm.)  13-15  16^18  19-21  22-24  25-27  28^30  31-33  34-36  37-39 
Number  1  22  52  67  114        257         177  41  2 


*  See  especially  J.  C.  Kapteyn,  Skew  frequency  curves  in  biology  and  statistics, 
Rec.  des  Trav.  Botan.  N4erland.,  Groningen,  xin,  pp.  105-158,  1916.  Also  Axel 
M.  Hemmingsen,  Statistical  analysis  of  the  differences  in  body-size  of  related  species, 
Danske  Vidensk.  Selsk.  Medd.  xcvm,  pp.  125-160,  1934. 

t  This  often  holds  good.  Wealth  breeds  wealth,  hence  the  distribution  of 
wealth  follows  a  skew  curve;  but  logarithmically  this  curve  becomes  a  normal 
one.  Weber's  law,  in  physiology,  is  a  well-known  instance;  on  the  thresholds 
of  sensations,  effects  are  produced  proportional  to  the  magnitudes  of  those 
thresholds,  and  the  logs  of  the  thresholds,  and  not  the  thresholds  themselves, 
are  normaUy  distributed. 


134 


THE  RATE  OF  GROWTH 


[CH. 


Let  us  sum  the  same  figures  up,  so  as  to  show  the  whole  number 
above  or  below  the  respective  sizes. 


Size  (mm.) 

15 

18 

21 

24 

27 

30 

33 

36 

39 

Number  below 

1 

23 

75 

142 

256 

513 

690 

731 

733 

Percentage 

— 

31 

10-2 

19-4 

34-9 

70-0 

941 

99-6 

100 

Our  first  set  of  figures,  the  actual  measurements,  would  give  us 
the  '*courbe  en  cloche,"  in  the  formrof  an  unsymmetrical  (or  "skew") 
Gaussian  curve:  one,  that  is  to  say,  with  a  long  sloping  talus  on 


Extreme 
Decile 

Quartile 


Median 


Quartile 

Decile 
Extreme 


Fig.  25  B. 


18 


33 


100% 
90 

75 


.50 


36       39 


21        24        27       30 
Length  in  mm. 
'Curve  of  distribution"  of  a  population  of  minnows. 


one  side  of  the  hilj.  The  other  gives  us  an  "S-shaped  curve,''  ap- 
parently hmited,  but  really  asymptotic  at  both  ends  (Fig.  25  B) ;  and 
this  S-shaped  curve  is  so  easy  to  work  with  that  we  may  at  once  divide 
it  into  two  halves  (so  finding  the  ''median"  value),  or  into  quarters 
and  tenths  (giving  the  "quartiles"  and  "deciles"),  or  as  we  please. 
In  short,  after  drawing  the  curVe  to  a  larger  scale,  we  shall  find  that 
we  can  safely  read  it  to  thirds  of  a  miUimetre,  and  so  draw  from  it 
the  following  somewhat  rough  but  very  useful  tabular  epitome  of 
our  population  of  minnows,  from  which  the  curve  can  be  recon- 
structed at  aiiy  time: 

mm. 

13 

210 

25-3 

28-6 

30-6 

32-3 


Extreme 
First  decile 
Lower  quartile 
Median 
Upper  quartile 
Last  decile 
Extreme 


39 


Ill] 


OF  Multimodal  curves 


135 


This  S-shaped  "summation-curve"  is  what  Francis  Galton  called  a 
curve  of  distribution,  and  he  "Uked  it  the  better  the  more  he  used  it." 
The  spread  or  "scatter"  is  conveniently  and  immediately  estimated 
by  the  distance  between  the  two  quartiles ;  and  it  happens  that  this 
very  nearly  coincides  with  the  standard  deviation  of  the  normal  curve. 


40 


50  60  70 

Micrometer-scale  units 


80 


90 


100 


Fig.  26.     A  plankton-sample  of  fish-eggs:  North  of  Scotland,  February  1905. 

(Only  eggs  without  oil-globule  are  counted  here.) 

A.  Dab  and  Flounder.     B,  Gadus  Esmarckii  and  G.  luscus. 

C,  Cod  and  Haddock.     D,  Plaice. 

There  are  biological  questions  for  which  we  want  all  the  accuracy 
which  biometric  science  can  give;  but  there  are  many  others  on 
which  such  refinements  are  thrown  away. 

Mathematically  speaking,  we  cannot  integrate  the  Gaussian  curve,, 
save  by  using  an  infinite  series;  but  to  all  intents  and  purposes  we 
are  doing  so,  graphically  and  very  easily,  in  the  illustration  we  have 
just  shewn.  In  any  case,  whatever  may  be  the  precise  character  of 
each,  we  begin  to  see  how  our  two  simplest  curves  of  growth,  the 
bell-shaped  and  the  S-shaped  curve,  form  a  reciprocal  pair,  the 
integral  and  the  differential  of  one  another  "^ — hke  the  distance  travelled 

*  It  is  of  considerable  historical  interest  to  know  that  this  practical  method  of 
summation  was  first  used  by  Edward  Wright,  in  a  Table  of  Latitudes  published  in 
his  Certain  Errors  in  Navigation  corrected,  1599,  as  a  means  of  virtually  integrating 
sec  X.  (On  this,  and  on  Wright's  claim  to  be  the  inventor  of  logarithms,  see  Florian 
Cajori,  in  Napier  Memorial  Volume,  1915,  pp.  94-99.) 


136  THE  RATE  OF  GROWTH  [ch. 

and  the  velocit}^  of  a  moving  body.     If  ^  =  e"^^  be  the  ordinate  of 
the  one,  z  =  le-^'^dx  is  that  of  the  other. 

There  is  one  more  kind  of  frequency-curve  which  we  must  take 
passing  note  of.  We  begin  by.  thinking  of  our  curve,  whether 
symmetrical  or  skew,  as  the  outcome  of  a  single  homogeneous 
group.  But  if  we  happen  to  have  two  distinct  but  intermingled 
groups  to  deal  with,  differing  by  ever  so  little  in  kind,  age,  place  or 
circumstance — leaves  of  both  oak  and  beech,  heights  of  both  men 
and  women — this  heterogeneity  will  tend  to  manifest  itself  in  two 
separate  cusps,  or  modes,  on  the  common  curve:  which  is  then 
indeed  two  curves  rolled  into  one,  each  keeping  something  of  its 
own  individuality.  For  example,  the  floating  eggs  of  the  food-fishes 
are  much  alike,  but  differ  appreciably  in  size.  A  random  gathering, 
netted  at  the  surface  of  the  sea,  will  yield  on  measurement  a  multi- 
modal curve,  each  cusp  of  which  is  recognisable,  more  or  less 
certainly,  as  belonging  to  a  particular  kind  of  fish  (Fig.  26). 

A  further  note  upon  curves 

A  statistical  "curve",  such  as  Quetelet  seems  to  have  been  the 
first  to  use*,  is  a  device  whose  peculiar  and  varied  beauty  we  are 
apt,  through  famiharity,  to  disregard.  The  curve  of  frequency  which 
we  have  been  studying  depicts  (as  a  rule)  the  distribution  of  mag- 
nitudes in  a  material  system  (a  population,  for  instance)  at  a 
certain  epoch  of  time ;  it  represents  a  given  state,  and  we  may  call 
/it  a  diagram  of  configuration  "f.  But  we  oftener  use  our  curves 
to  compare  successive  states,  or  changes  of  magnitude,  as  one 
configuration  gives  place  to  another;  and  such  a  curve  may  be 
called  a  diagram  of  displacement.  An  imaginary  point  moves  in 
imaginary  space,  the  dimensions  of  which  represent  those  of  the 
phenomenon  in  question,  dimensions  which  we  may  further  define 
and  measure  by  a  system  of  "coordinates";  the  movements  of  our 
point  through  its  figurative  space  are  thus  analogous  to,  and  illus- 
trative of,  the  events  which  constitute  the  phenomenon.  Time  is 
often  represented,  and  measured,  on  one  of  the  coordinate  axes,  and 
our  diagram  of  "displacement"  then  becomes  a  diagram  of  velocity. 

*  In  his  Theorie  des  probabilites,  1846. 

t  See  Clerk  Maxwell's  article  "Diagrams,"  in  the  Encyclopaedia  Britannica, 
9th  edition. 


Ill]  OF  STATISTICAL  CURVES  137 

This  simple  method  (said  Kelvin)  of  shewing  to  the  eye  the  law  of 
variation,  however  complicated,  of  an  independent  variable,  is  one 
of  the  most  beautiful  results  of  mathematics*. 

We  make  and  use  our  curves  in  various  ways.  We  set  down  on 
the  coordinate  network  of  our  chart  the  points  givQ^i  by  a  series  of 
observations,  and  connect  them  up  into  a  continuous  series  as  we 
chart  the  voyage  of  a  ship  from  her  positions  day  by  day ;  we  may 
"smooth"  the  line,  if  we  so  desire.  Sometimes  we  find  our  points 
so  crowded,  or  otherwise  so  dispersed  and  distributed,  that  a  line 
can  be  drawn  not  from  one  to  another  but  among  them  all — a  method 
first  used  by  Sir  John  Herschel  f,  when  he  studied  the  orbits  of  the 
double  stars.  His  dehcate  observations  were  affected  by  errors,  at 
first  sight  without  rhyme  or  reason,  but  a  curve  drawn  where  the 
points  lay  thickest  embodied  the  common  lesson  of  them  all;  any 
one  pair  of  observations  would  have  sufiiced,  whether  better  or 
worse,  for  the  calculation  of  an  orbit,  but  Herschel's  dot-diagram 
obtained  "from  the  whole  assemblage  of  observations  taken  together, 
and  regarded  as  a  single  set  of  data,  a  single  result  in  whose  favour 
they  all  conspire."  It  put  us  in  possession,  said  Herschel,  of 
something  truer  than  the  observations  themselves  % ;  and  Whewell 
remarked  that  it  enabled  us  to  obtain  laws  of  Nature  not  only  from 
good  but  from  very  imperfect  observations  §.  These  are  some 
advantages  of  the  use  of  "curves,"  which  have  made  them  essential 
to  research  and  discovery. 

It  is  often  helpful  and  sometimes  necessary  to  smooth  our  curves, 

*  Kelvin,  Nature,  xxix,  p.  440,  1884. 

t  Mem.  Astron.  Soc.  v,  p.  171,  1830;  Nautical  Almanack,  1835,  p.  495;  etc. 

X  Here  a  certain  distinction  may  be  observed.  We  take  the  average  height  of  a 
regiment,  because  the  men  actually  vary  about  a  mean.  But  in  estimating  the  place 
of  a  star,  or  the  height  of  Mont  Blanc,  we  average  results  which  only  differ  by 
I)ersonal  or  instrumental  error.  It  is  this  latter  process  of  averaging  which  leads, 
in  Herschel's  phrase,  to  results  more  trustworthy  than  observation  itself.  Laplace 
had  made  a  similar  remark  long  before  {Oeuvres,yii,  Theorie  des  probabilites) :  that 
we  may  ascertain  the  very  small  effect  of  a  constant  cause,  by  means  of  a  long  series 
of  observations  the  errors  of  which  exceed  the  effect  itself.  He  instances  the  small 
deviation  to  the  eastward  which  the  rotation  of  the  earth  imposes  on  a  falling  body. 
In  like  manner  the  mean  level  of  the  sea  may  be  determined  to  the  second  decimal 
of  an  inch  by  observations  of  high  and  low  water  taken  roughly  to  the  nearest  inch, 
provided  these  are  faithfxilly  carried  out  at  every  tide,  for  say  a  hundred  years. 
Cf.  my  paper  on  Mean  Sea  Level,  in  Scottish  Fishery  Board's  Sci.  Report  for  1915. 

§  Novum  Organum  Renovatum  (3rd  ed.),  1858,  p.  20. 


138  THE  RATE  OF  GROWTH  [ch. 

whether  at  free  hand  or  by  help  of  m^athematical  rules;  it  is  one 
way  of  getting  rid  of  non-essentials — and  to  do  so  has  been  called 
the  very  key-note  of  mathematics*.  A  simple  rule,  first  used  by 
Gauss,  is  to  replace  each  point  by  a  mean  between  it  and  its  two 
or  more  neighbours,  and  so  to  take  a  "floating"  or  "running 
average."  In  so  doing  we  trade  once  more  on  the  "principle  of 
continuity";  and  recognise  that  in  a  series  of  observations  each 
one  is  related  to  another,  and  is  part  of  the  contributory  evidence 
on  which  our  knowledge  of  all  the  rest  depends.  But  all  the  while 
we  feel  that  Gaussian  smoothing  gives  us  a  practical  or  descriptive 
result,  rather  than  a  mathematical  one. 

Some  curves  are  more  elegant  than  others.  We  may  have  to  rest 
content  with  points  in  which  no. order  is  apparent,  as  when  we  plot 
the  daily  rainfall  for  a  month  or  two;  for  this  phenomenon  is  one 
whose  regularity  only  becomes  apparent  over  long  periods,  when 
average  values  lead  at  last  to  "statistical  uniformity."  But  the 
most  irregular  of  curves  may  be  instructive  if  it  coincide  with  another 
not  less  irregular :  as  when  the  curve  of  a  nation's  birth-rate,  in  its 
ups  and  downs,  follows  or  seems  to  follow  the  price  of  wheat  or  the 
spots  upon  the  sun. 

It  seldom  happens,  outside  of  the  exact  sciences,  that  we  com- 
prehend the  mathematical  aspect  of  a  phenomenon  enough  to  define 
(by  formulae  and  constants)  the  curve  which  illustrates  it.  But, 
failing  such  thorough  comprehension,  we  can  at  least  speak  of  the 
trend  of  our  curves  and  put  into  words  the  character  and  the  course 
of  the  phenomena  they  indicate.  We  see  how  this  curve  or  that 
indicates  a  uniform  velocity,  a  tendency  towards  acceleration  or 
retardation,  a  periodic  or  non-periodic  fluctuation,  a  start  from  or  an 
approach  to  a  limit.  When  the  curve  becomes,  or  approximates  to, 
a  mathematical  one,  the  types  are  few  to  which  it  is  Kkely  to 
belong f.  A  straight  line,  a  parabola,  or  hyperbola,  an  exponential 
or  a  logarithmic  curve  (like  x'=ay^),  a  sine-curve  or  sinusoid,  damped 
or  no,  suffice  for  a  wide  range  of  phenomena;  we  merely  modify  our 
scale,  and  change  the  names  of  our  coordinates. 

*  Cf.  W.  H.  Young,  The  mathematic  method  and  its  limitations,  Atti  del  Congresso 
dei  Matematici,  Bologna,  192/8,  i,  p.  203. 

t  Hence  the  engineer  usually  begins,  for  his  first  tentative  construction,  by 
drawing  one  of  the  familiar  curves,  catenary,  parabola,  arc  of  a  circle,  or  curve  of, 
sines. 


Ill]  OF  STATISTICAL  CURVES  139 

The  curves  we  mostly  use,  other  than  the  Gaussian  curve,  are 
time-diagrams.  Each  has  a  beginning  and  an  end;  and  one  and 
the  same  curve  may  illustrate  the  life  of  a  man,  the  economic  history 
of  a  kingdom,  the  schedule  of  a  train  between  one  st^^tion  and 
another.  What  it  then  shews  is  a  velocity,  an  acceleration,  and 
a  subsequent  negative  acceleration  or  retardation.  It  depicts  a 
"mechanism"  at  work,  and  helps  us  to  see  analogous ' mechanisms 
in  different  fields;  for  Nature  rings  her  many  changes  on  a  few 
simple  themes.  The  same  expressions  serve  for  different  orders  of 
phenomena.  The  swing  of  a  pendulum,  the  flow  of  a  current,  the 
attraction  of  a  magnet,  the  shock  of  a  blow,  have  their  analogues  in 
a  fluctuation  of  trade,  a  wave  of  prosperity,  a  blow  to  credit,  a  tide 
in  the  affairs  of  men. 

The  same  exponential  curve  may  illustrate  a  rate  of  cooHng,  a  loss 
of  electric  charge,  the  chemical  action  of  a  ferment  or  a  catalyst. 
The  S-shaped  population-curve  or  "logistic  curve"  of  Verhulst  (to 
which  we  are  soon  coming)  is  the  hysteresis-curve  by  which  Ewing 
represented  self-induction  in  a  magnetic  field ;  it  is  akin  to  the  path 
of  a  falhng  body  under  the  influence  of  friction;  and  Lotka  has 
drawn  a  curve  of  the  growing  mileage  of  American  railways,  and 
found  it  to  be  a  typical  logistic  curve.  A  few  bars  of  music  plotted 
in  wave-lengths  of  the  notes  might  be  mistaken  for  a  tidal  record. 
The  periodicity  of  a  wave,  the  acceleration  of  gravity,  retardation 
by  friction,  the  role  of  inertia,  the  explosive  action  of  a  spark  or 
an  electric  contact — these  are  some  of  the  modes  of  action  or  "forms 
of  mechanism"  which  recur  in  Hmited  number,  but  in  endless  shapes 
and  circumstances*.  The  way  in  which  one  curve  fits  many 
phenomena  is  characteristic  of  mathematics  itself,  which  does  not  deal 
with  the  specific  or  individual  case,  but  generalises  all  the  while,  and 
is  fond  (as  Henri  Poincare  said)  of  giving  the  same  name  to  different 
things. 

Our  curves,  as  we  have  said,  are  mostly  time-diagrams,  and 
represent  a  change  in  time  from  one  magnitude  to  another;  they  are 
diagrams  of  displacement,  in  Maxwell's  phrase.  We  may  consider 
four  different  cases,  not  equally  simple   mathematically,  but  all 

*  See  an  admirable  little  book  by  Michael  Petrovich,  Les  micanismes  communs 
aux  phenomenes  disparates,  Paris,  1921. 


140  THE  RATE  OF  GROWTH  [ch. 

capable   of  explanation,   up   to   a  certain  point,  without   mathe- 
matics. 

(1)  If  in  our  coordinate  diagram  we  have  merely  to  pass  from 
one  isolated  jpoint  to  another,  a  straight  line  joining  the  two  points 
is  the  shortest — and  the  hkeliest  way. 

(2)  To  rise  and  fall  alternately,  going  to  and  fro  from  maximum 
to  minimum,  a  zig-zag  rectilinear  path  would  still  be,  geometrically, 
the  shortest  way;  but  it  would  be  sharply  discontinuous  at  every 
turn,  it  would  run  counter  to  the  "principle  of  continuity,"  it  is  not 
likely  to  be  nature's  way.  A  wavy  course,  with  no  more  change  of 
curvature  than  is  absolutely  necessary,  is  the  path  which  nature 
follows.     We  call  it  a  simple  harmonic  motion,  and  the  simplest  of 


The  Sine -curve 


The  S- shaped  curve 


The  bell-shaped  curve 
Fig.  27.     Simple  curves,  representing  a  change  from  one  magnitude  to  another. 

all  such  wavy  curves  we  call  a  sine-curve.  If  there  be  but  one 
maximum  and  one  minimum,  which  our  variant  alternates  between, 
the  vector  pathway  may  be  translated  into  jpolar  coordinates;  the 
vector  does-  what  the  hands  of  the  clock  do,  and  a  circle  takes  the 
place  of  the  sine-curve. 

(3)  To  pass  from  a  zero-line  to  a  maximum  once  for  all  is  a  very 
different  thing ;  for  now  minimum  and  maximum  are  both  of  them 
continuous  states,  and  the  principle  of  continuity  will  cause  our 
vector-variant  to  leave  the  one  gradually,  and  arrive  gradually  at 
the  other.  The  problem  is  how  to  go  uphill  from  one  level  road  to 
another,  with  the  least  possible  interruption  or  discontinuity.  The 
path  follows  an  S-shaped  course;  it  has  an  inflection  midway;  and 
the  first  phase  and  the  last  are  represented  by  horizontal  asymptotes. 
This  is  an  important  curve,  and  a  common  one.     It  so  far  resembles 


Ill]  OF  CURVES  IN  GENERAL  141 

an  "elastic  curve"  (though  it  is  not  mathematically  identical  with  it) 
that  it  may  be  roughly  simulated  by  a  watchspring,  lying  between 
two  parallel  straight  lines  and  touching  both  of  them.  It  has  its 
kinetic  analogue  in  the  motion  of  a  pendulum,  which  starts  from 
rest  and  comes  to  rest  again,  after  passing  midway  through  its 
maximal  velocity.  It  indicates  a  balance  between  production  and 
waste,  between  growth  and  decay:  an  approach  on  either  side  to 
a  state  of  rest  and  equilibrium.  It  shows  the  speed  of  a  train 
between  two  stations ;  it  illustrates  the  growth  of  a  simple  organism, 
or  even  of  a  population  of  men.  A  certain  simple  and  symmetrical 
case  is  called  the  Verhulst- Pearl  curve,  or  the  logistic  curve. 

(4)  Lastly,  in  order  to  leave  a  certain  minimum,  or  zero-hne,  and 
return  to  it  again,  the  simplest  way  will  be  by  a  curve  asymptotic 
to  the  base-line  at  both  ends — or  rather  in  both  directions;  it  will 
be  a  bell-shaped  curve,  having  a  maximum  midway,  and  of  necessity 
a  point  of  inflection  on  either  side ;  it  is  akin  to,  and  under  certain 
precise  conditions  it  becomes,  the  curve  of  error  or  Gaussian  curve. 

Besides  the  ordinary  curve  of  growth,  which  is  a  summation- 
curve,  and  the  curve  of  growth-rates,  which  is  its  derivative,  there 
are  yet  others  which  we  may  employ.  One  of  these  was  introduced 
by  Minot*,  from 'a  feeling  that  the  rate  of  growth,  or  the  amount 
of  increment,  ought  in  some  way  to  be  equated  with  the  growing 
structure.  Minot's  method  is  to  deal,  not  with  the  actual  increments 
added  in  successive  periods,  but  with  these  successive  increments 
represented  as  percentages  of  the  amount  already  reached.  For 
instance,  taking  Quetelet's  values  for  the  height  (in  centimetres)  of 
a  male  infant,  we  have  as  follows: 


Years 

0 

1 

2 

3 

4 

era. 

500 

69-8 

791 

86-4 

92-7 

But  Minot  would  state  the  percentage-growth  in  each  of  these 
four  annual  periods  at  39-6,  13-3,  9-2  and  7-3  per  cent,  respectively: 

Years       0  1  2  3  4 

Height  (cm.)  50-0  69-8  79-1  86-4  92-7 

Increments  (cm.)  —  19-8  9-3  7-3  6-3 

(per  cent.)  ~  39-6  13-3  9-2  7-3 

*  C.  S.  Minot,  On  certain  phenomena  of  growing  old,  Proc.  Amer.  Assoc,  xxxix, 
1890,  21  pp.;  Senescence  and  rejuvenation,  Journ.  Physiol,  xii,  pp.  97-153, 
1891;   etc.     Criticised  by  S.  Brody  and  J.  Needham,  ojp,  cit.  pp.  401  seq. 


142  THE  RATE  OF  GROWTH  [ch. 

Now,  in  our  first  curve  of  growth  we  plotted  length  against  time, 
a  very  simple  thing  to  do.  When  we  differentiate  L  with  respect  to  T, 
we  have  dL/dT,  which  is  rate  or  velocity,  again  a  very  simple  thing ; 
and  from  this,  by  a  second  differentiation,  we  obtain,  if  necessary, 
d^L/dT^,  that  is  to  say,  the  acceleration. 

But  when  you  take  percentages  of  y,  you  are  determining  dyjy, 
and  w^hen  you  plot  this  against  dx,  you  have 

^,     or  -^,     or  -.^. 
dx  '  y.dx'  y' dx' 

That  is  to  say,  you  are  multiplying  the  thing  whose  variations 
you  are  studying  by  another  quantity  which  is  itself  continually 
varying;  and  are  dealing  with  something  more  complex  than  the 
original  factors*.  Minot's  method  deals  with  a  perfectly  legitimate 
function  of  x  and  y,  and  is  tantamount  to  plotting  log  y  against  x, 
that  is  to  say,  the  logarithm  of  the  increment  against  the  time. 
This  would  be  all  to  the  good  if  it  led  to  some  simple  result,  a  straight 
line  for  instance ;  but  it  is  seldom  if  ever,  as  it  seems  to  me,  that  it 
does  anything  of  the  kind.  It  has  also  been  pointed  out  as  a  grave 
fault  in  his  method  that,  whereas  growth  is  a  continuous  process, 
Minot  chooses  an  arbitrary  time-interval  as  his  basis  of  comparison, 
and  uses  the  same  interval  in  all  stages  of  development.  There  is 
little  use  in  comparing  the  percentage  increase  fer  week  of  a  week- 
old  chick,  with  that  of  the  same  bird  at  six  months  old  or  at  six 
years. 


The  growth  of  a  population 

After  dealing  with  Man's  growth  and  stature,  Quetelet  turned  to 
the  analogous  problem  of  the  growth  of  a  populatfon — all  the  more 
analogous  in  our  eyes  since  we  know  man  himself  to  be  a  "statistical 
unit,"  an  assemblage  of  organs,  a  population  of  cells.     He  had  read 

*  Schmalhausen,  among  others,  uses  the  same  measure  of  rate  of  growth,  in  the 
form 

log  F-logF      dvl^ 
^~      k{t-t)      ~^  dt    v' 

Arch.  f.  Entw.  Mech.  cxm,  pp.  462-519,  1928. 


m]  MALTHUS  ON  POPULATION  143 

Malthus's  Essay  on  Population*  in  a  French  translation,  and  was 
impressed  like  all  the  world  by  the  importance  of  the  theme.  He 
saw  that  poverty  and  misery  ensue  when  a  population  outgrows  its 
means  of  support,  and  believed  that  multiphcation  is  checked  both 
.  by  lack  of  food  and  fear  of  poverty.  He  knew  that  there  were, 
and  must  be,  obstacles  of  one  kind  or  another  to  the  unrestricted 
increase  of  a  population;  and  he  knew  the  more  subtle  fact  that 
a  population,  after  growing  to  a  certain  height,  oscillates  about  an 
unstable  level  of  equilibrium  f. 

Malthus  had  said  that  a  population  grows  by  geometrical  pro- 
gression (as  1,  2,  4,  8)  while  its  means  of  subsistence  tend  rather  to 
grow  by  arithmetical  (as  1,  2,  3,  4) — that  one  adds  up  while  the 
other  multiphesj.     A  geometrical  progression  is  a  natural  and  a 

*  T.  R.  Malthus,  An  Essay  on  the  Principle  of  Population,  as  it  affects  the  Future 
Improvement  of  Society,  etc.,  1798  (6th  ed.  1826;  transl.  by  P.  and  G.  Prevost, 
Geneva,  1830,  1845).  Among  the  books  to  which  Malthus  was  most  indebted  was 
A  Dissertation  on  the  Numbers  of  Mankind  in  ancient  and  modern  Times,  published 
anonymously  in  Edinburgh  in  1753,  but  known  to  be  by  Robert  Wallace  and  read 
by  him  some  years  before  to  the  Philosophical  Society  at  Edinburgh.  In  this 
remarkable  work  the  writer  says  (after  the  manner  of  Malthus)  that  mankind 
naturally  increase  by  successive  doubling,  and  tend  to  do  so  thrice  in  a  hundred 
years.  He  explains,  on  the  other  hand,  that  "mankind  do  not  actually  propagate 
according  to  the  rule  in  our  tables,  or  any  other  constant  rule;  ^et  tables  of  this 
nature  are  not  entirely  useless,  but  may  serve  to  shew,  how  much  the  increase  of 
mankind  is  prevented  by  the  various  causes  which  confine  their  number  within 
such  narrow  limits."  Malthus  was  also  indebted  to  David  Hume's  Political 
Discourse,  Of  the  Populousness  of  ancient  Nations,  1752,  a  work  criticised  by  Wallace. 
See  also  McCulloch's  notes  to  Adam  Smith's  Wealth  of  Nations,  1828. 

■f"  That  the  nearest  approach  to  equilibrium  in  a  population  is  long- continued 
ebb  and  flow,  a  mean  level  and  a  tide,  was  known  to  Herbert  Spencer,  and  was 
stated  mathematically  long  afterwards  by  Vito  Volterra.  See  also  Spencer's  First 
Principles,  ch. ^22,  sect.  173:  "Every  species  of  plant  or  animal  is  perpetually 
undergoing  a  rhythmical  variation  in  number — now  from  abundance  of  food  and 
absence  of  enemies  rising  above  its  average,  and  then  by  a  consequent  scarcity 
of  food  and  abundance  of  enemies  being  depressed  below  its  average.. .  .Amid 
these  oscillations  produced  by  their  conflict,  lies  that  average  number  of  the  species 
at  which  its  expansive  tendency  ie  in  equilibrium  with  surrounding  repressive 
tendencies."  Cf.  A.  J.  Lotka,  Analytical  note  on  certain  rhythmic  relations  in 
organic  systems,  Proc.  Nat.  Acad.  Sci.  vi,  pp.  ^10-415,  1920;  but  cf.  also  his  Elements 
of  Physical  Biology,  1915,  p.  90.  An  analogy,  and  perhaps  a  close  one,  may  be  found 
on  the  Bourse  or  money  market. 

X  That  a  population  will  soon  oiitrun  its  means  of  subsistence  was  a  natural 
assumption  in  Malthus's  day,  and  in  his  own  thickly  populated  land.  The  danger 
may  be  postponed  and  the  assumption  apparently  falsified,  as  by  an  Argentine 
cattle-ranch  /or  prairie  wheat-farm — but  only  so  long  as  we  enjoy  world-wide 
freedom  of  import  and  exchange. 


144  THE  RATE  OF  GROWTH  [ch. 

common  thing,  and,  apart  from  the  free  growth  of  a  population  or 
an  organism,  we  find  it  in  many  biological  phenomena.  An  epidemic 
dechnes,  or  tends  to  decline,  at  a  rate  corresponding  to  a  geometrical 
progression;  the  mortahty  from  zymotic  diseases  declines  in  geo- 
metrical progression  among  children  from  one  to  ten  years  old; 
and  the  chances  of  death  increase  in  geometrical  progression  after 
a  certain  time  of  Hfe  for  us  all*. 

But  in  the  ascending  scale,  the  story  of  the  horseshoe  nails  tells 
us  how  formidable  a  thing  successive  multipHcation  becomes  f. 
Enghsh  law  forbids  the  protracted  accumulation  of  compound 
interest;  and  hkewise  Nature  deals  after  her  own  fashion  with  the  case, 
and  provides  her  automatic  remedies.  A  fungus  is  growing  on  an 
oaktree — it  sheds  more  spores  in  a  night  than  the  tree  drops  acorns 
in  a  hundred  years.  A  certain  bacillus  grows  up  and  multiphes  by 
two  in  two  hoTlrs'  time;  its  descendants,  did  they  all  survive,  would 
number  four  thousand  in  a  day,  as  a  man's  might  in  three  hundred 
years.  A  codfish  lays  a  million  eggs  and  more — all  in  order  that 
one  pair  may  survive  to  take  their  parents'  places  in  the  world. 
On  the  other  hand,  the  humming-birds  lay  only  two  eggs,  the  auks 
and  guillemots  only  one;  yet  the  former  are  multitudinous  in  their 
haunts,  and  some  say  that  the  Arctic  auks  and  auklets  outnumber 
all  other  birds  in  the  world.  Linnaeus  {  shewed  that  an  annual 
plant  would  have  a  miUion  offspring  in  twenty  years,  if  only  two 
seeds  grew  up  to  maturity  in  a  year. 

But  multiply  as  they  will,  these  vast  populations  have  their 
limits.  They  reach  the  end  of  their  tether,  the  pace  slows  down,  and 
at  last  they  increase  no  more.  Their  world  is  fully  peopled,  whether 
it  be  an  island  with  its  swarms  of  humming-birds,  a  test-tube  with 
its  myriads  of  yeast-cells,  or  a  continent  with  its  miUions  of  mankind. 
Growth,  whether  of  a  population  or  an  individual,  draws  to  its 
natural  end;  and  Quetelet  compares  it,  by  a  bold  metaphor,  to  the 
motion  of  a  body  in  a  resistant  medium.  A  typical  population 
grows  slowly  from  an  asymptotic  minimum;  it  multiphes  quickly; 

*  According  to  the  Law  of  Gompertz ;  cf.  John  Brownlee,  in  Proc.  R.S.E.  xxxi, 
pp.  627-634,  1"911. 

t  Herbert  Spencer,  A  theory  of  population  deduced  from  the  general  law  of 
animal  fertility,  Westminster  Review,  April  1852. 

X  In  his  essay  De  Tellure,  1740. 


Ill]  VERHULST'S  LAW  145 

it  draws  slowly  to  an  ill-defined  and  asymptotic  maximum.  The 
two  ends  of  the  population-curve  define,  in  a  general  way,  the 
whole  curve  between;  for  so  beginning  and  so  ending  the  curve 
must  pass  through  a  point  of  inflection,  it  must  be  an  S-shaped 
curve.  It  is  just  such  a  curve  as  we  have  seen  imder  simple 
conditions  of  growth  in  an  individual  organism. 

This  general  and  all  but  obvious  trend  of  a  population-curve  has 
been  recognised,  with  more  or  less  precision,  by  many  writers.  It 
is  imphcit  in  Quetelet's  own  words,  as  follows:  "Quand  une 
population  pent  se  developper  hbrement  et  sans  obstacles,  elle  croit 
selon  une  progression  geometrique;  si  le  developpement  a  heu  au 
miheu  d'obstacles  de  toute  espece  qui  tendent  a  Farreter,  et  qui 
agissent  d'une  maniere  uniforme,  c'est  a  dire  si  I'etat  sociale  ne 
change  point,  la  population  n'augmente  pas  d'une  maniere  indefinie, 
mais  elle  tend  de  plus  en  plus  a  devenir  stationnaire*.''  P.  F.  Verhulst, 
a  mathematical  colleague  of  Quetelet's,  was  interested  in  the  same 
things,  and  tried  to  give  a  mathematical  shape  to  the  same  general 
conclusions;  that  is  to  say,  he  looked  for  a  "fonction  retardatrice" 
which  should  turn  the  Malthusian  curve  of  geometrical  progression 
into  the  S-shaped,  or  as  he  called  it,  the  logistic  curve,  which  should 
thus  constitute  the  true  "law  of  population,"  and  thereby  indicate 
(among  other  things)  the  hmit  above  which  the  population^  was  not 
likely  to  grow  f . 

Verhulst  soon  saw  that  he  could  only  solve  his  problem  in  a 
prehminary  and  tentative  way;  ''la  hi  de  la  population  nous  est 
inconnue,  parcequ'on  ignore  la  nature  de  la  fonction  qui  sert  de 
mesure  aux  obstacles  qui  s'opposent  a  la  multiphcation  indefini  de 
I'espece  humaine."  The  materials  at  hand  were  almost  unbeUevably 
scanty  and  poor.  The  French  statistics  were  taken  from  documents 
"qui  ont  ete  reconnus  entierement  fictifs";  in  England  the  growth 

*  Physique  Sociale,  i,  p.  27,  1835.  But  Quetelet's  brief  account  is  somewhat 
ambiguous,  and  he  had  in  mind  a  body  falling  through  a  resistant  medium — which 
suggests  a  limiting  velocity,  or  limiting  annual  increment,  rather  than  a  terminal 
value.  See  Sir  G.  Udny  Yule,  The  growth  of  population,  Journ.  R.  Statist.  Soc. 
Lxxxvm,  p.  42,  1925. 

t  P.  F.  Verhulst,  Notice  sur  la  loi  que  la  population  suit  dans  son  accroissement. 
Correspondence  math.  etc.  public  par  M.  A.  Quetelet,  x,  pp.  113-121,  1838;  Rech. 
math,  sur  la  loi  etc.,  Nozw.  Mem.  de  VAcad.  R.  de  Bruxelles,  xviii,  38  pp.,  1845; 
deuxieme  Mem.,  ihid.  xx,  32  pp.,  1847.  The  term  logistic  curve  had  already  been 
used  by  Edward  Wright;  see  antea,  p.  135.  footnote. 


146  THE  RATE  OF  GROWTH  [ch. 

of  the  population  was  estimated  by  the  number  of  births,  and  the 
births  by  the  baptisms  in  the  Church  of  England,  "de  maniere  que 
les  enfants  des  dissidents  ne  sont  point  portes  sur  les  registres 
officiels."  A  law  of  population,  or  "loi  d'affaibhssement"  became 
a  mere  matter  of  conjecture,  and  the  simplest  hypothesis  seemed  to 
Verhulst  to  be,  to  regard  "cet  affaibhssement  comme  proportionnel 
a  I'accroissement  de  la  population,  depuis  le  moment  ou  la  difficulty 
de  trouver  de  bonnes  terres  a  commence  a  se  faire  sentir*." 

Verhulst  was  making  two  assumptions.  The  first,  which  is  beyond 
question,  is  that  the  rate  of  increase  cannot  be,  and  indeed  is  not, 
a  constant;  and  the  second  is  that  the  rate  must  somehow  depend 
on  (or  be  some  function  of)  the  population  for  the  time  being. 
A  third  assumption,  again  beyond  question,  is  that  the  simplest 
possible  function  is  a  linear  function.  He  suggested  as  the  simplest 
possible  case  that,  once  the  rate  begins  to  fall  (or  once  the  struggle 
for  existence  sets  in),  it  will  fall  the  more  as  the  population  continues 
to  grow;  we  shall  have  a  growth-factor  and  a  retardation-factor  in 
proportion  to  one  another.  He  was  making  early  use  of  a  simple 
differential  equation  such  as  Vito  Volterra  and  others  now  employ 
freely  in  the  general  study  of  natural  selec.tionf. 

The  point  wher^  a  struggle  for  existence  first  sets  in,  and  where 
ipso  facto  the  rate  of  increase  begins  to  diminish,  is  called  by  Verhulst 
the  normal  level  of  the  population ;  he  chooses  it  for  the  origin  of  his 
curve,  which  is  so  defined  as  to  be  symmetrical  on  either  side  of 
this  origin.  Thus  Verhulst's  law,  and  his  logistic  curve,  owe  their 
form  and  their  precision  and  all  their  power  to  forecast  the  future 
to  certain  hypothetical  assumptions;  and  the  tentative  solution 
arrived  at  is  one  "sous  le  point  de  vue  mathematiquej." 

♦  Op.  cit.  p.  8. 

t  Besides  many  well-known  papers  by  Volterra,  see  V.  A,  Kostitzin,  Biologie 
mathematique,  Paris,  1937.  Cf.  also,  for  the  so-called  "Malaria  equations,"  Ronald 
Ross,  Prevention  of  Malaria,  2nd  ed.  1911,  p.  679;  Martini,  Zur  Epidemiologie  d. 
Malaria,  Hamburg,  1921;  W.  R.  Thompson,  C.R.  clxxiv,  p.  1443,  1922,  C.  N. 
Watson,  Nature,  cxi,  p.  88,  1923. 

J  Verhulst  goes  on  to  say  that  "  une  longue  serie  d 'observations,  non  interrompues 
par  de  grandes  catastrophes  sociales  ou  des  revolutions  du  globe,  fera  probablement 
decouvrir  la  fonction  retardatrice  dont  il  vient  d'etre  fait  mention."  Verhulst 
simplified  his  problem  to  the  utmost,  but  it  is  more  complicated  today  than  ever; 
he  thought  it  impossible  that  a  country  should  draw  its  bread  and  meat  from 
overseas:    "lors  meme  qu'une  partie  considerable  de  la  population  pourrait  etre 


Ill]  THE  LOGISTIC  CURVE  147 

The  mathematics  of  the  Verhulst-Pearl  curve  need  hardly  concern 
us;  they  are  fully  dealt  with  in  Raymond  Pearl's,  Lotka's  and  other 
books.  Verhulst  starts,  as  Malthus  does,  with  a  population  growing 
in  geometrical  progression,  and  so  giving  a  logarithmic  curve: 

dp 

He  then  assumes,  as  his  "loi  d'affaibUssement,"  a  coefficient  of 
retardation  {n)  which  increases  as  the  population  increases: 


dp  2 


Integrating,  p  =  ^^_^_^_. 

If  the  point  of  inflection  be  taken  as  the  origin,  k  =  0;  and  again 
for  <  =  00,  jt?  =  —  =  L.     We  may  write  accordingly: 


1  +  e-^ 


Malthus  had  reckoned  on  a  population  doubling  itself,  if  unchecked 
by  want  or  "accident,"  every  twenty-five  years*;  but  fifty  years 
after,  Verhulst  shewed  that  this  "grande  vitesse  d'accroissement" 
was  no  longer  to  be  found  in  France  or  Belgium  or  other  of  the 
older  countries!,  but  wa^  still  being  reaUsed  in  the  United  States 
(Fig.  28).  All  over  Europe,  "le  rapport  de  I'exces  annuel  des  nais- 
sances  sur  les  deces,  a  la  population  qui  I'a  fourni,  va  sans  cesse  en 
s'affaiblissant ;  de  maniere  que  Faccroissement  annuel,  dont  la  valeur 
absolue  augmente  continuellement  lorsqu'il  y  a  progression  geo- 
metrique,  parait  suivre  une  progression  tout  au  plus  arithmetique." 

nourrie  de  bles  etrangers,  jamais  un  gouvernement  sage  ne  consentira  a  faire 
dependre  I'existence  de  milliers  de  citoyens  du  bon  vouloir  des  souverains  etrangers." 
On  this  and  other  problems  in  the  growth  of  a  human  population,  see  L.  Hogben's 
Genetic  Problems,  etc.,  1937,  chap.  vii.  See^also  [int.  al.)  Warren  S.  Thompson  and 
P.  K.  Whelpton,  Population  Trends  in  the  United  States,  1933;  F.  Lorimer  and 
F.  Osborn,  Dynamics  of  Population,  1934,  etc. 

*  An  estimate  based,  like  the  rest  of  Malthus's  arithmetic,  on  very  slender 
evidence. 

t  In  Quetelet's  time  the  European  countries,  far  from  doubling  in  twenty-five 
years,  were  estimated  to  do  so  in  from  sixty  years  (Norway)  to  four  hundred  years 
(France);  see  M.  Haushofer,  Lehrbuch  der  Statiatik,  1882. 


148 


THE  RATE  OF  GROWTH 


[CH. 


The  "celebrated  aphorism"  of  Malthus  was  thus,  and  to  this  extent, 
confirmed*.  In  the  United  States,  the  Malthusian  estimate  of 
unrestricted  increase  continued  to  be  reahsed  for  a  hundred  years 
after  Malthus  wrote;  for  the  3-93  millions  of  the  U.S.  census  of 
1790  were  doubled  three  times  over  in  the  census  of  1860,  and  four 
times  over  in  that  of  1890.  A  capital  which  doubles  in  twenty-five 
years  has  grown  at  2-85  per  cent,  per  annum,  compound  interest; 
the  U.S.  population  did  rather  more,  for  it  grew  at  fully  3  per  cent, 
for  fifty  of  those  hundred  years  f. 


1790  1800 


1850 


1900 


1930 


Fig.  28.     Population  of  the  United  States,  1790-1930. 

The  population  of  the  whole  world  and  of  every  continent  has 
increased  during  modem  times,  and  the  increase  is  large  though 
the  rate  is  low.  The  rate  of  increase  has  been  put  at  about  half-a- 
per-cent  per  annum  for  the  last  three  hundred  years — a  shade  more 
in  Europe  and  a  shade  less  in  the  rest  of  the  world  { : 

*  Op.  cit.  1845,  p.  7. 

t  Verhulst  foretold  forty  millions  as  the  "extreme  limit"  of  the  population 
of  France,  and  6^  millions  as  that  of  Belgium.  The  latter  estimate  he  increased 
to  8  millions  later  on.  The  actual  populations  of  France  and  Belgium  at  the 
present  time  are  a  little  more  than  the  ultimate  limit  which  Verhulst  foretold. 

+  From  A.  M.  Carr-Saunders'  World  Population,  1936,  p.  30. 


in]  RAYMOND  PEARL  149 

An  estimate  of  the  population  of  the  world 
(After  W.  F.  Willcox) 

Mean  rate  of 
1650         1750        1800         1850         1900  increase 

Europe  100  140  187  266  401         0-52  %  per  annum 

World  total        545  728  906         1171         1608         0-49%    „ 

Verhulst  was  before  his  time,  and  his  work  was  neglected  and 
presently  forgotten.  Only  some  twenty  years  ago,  Raymond  Pearl 
and  L.  J.  Reed  of  Baltimore,  studying  the  U.S.  population  as 
Verhulst  had  done,  approached  the  subject  in  the  same  way,  and 
came  to  an  identical  result;  then,  soon  afterwards  (about  1924), 
Raymond  Pearl  came  across  Verhulst's  papers,  and  drew  attention 
to  what  we  now  speak  of  as  the  Verhulst-Pearl  law.  Pearl  and 
Reed  saw,  as  Verhulst  had  dope,  that  a  "law  of  population"  which 
should  cover  all  the  ups  and  downs  of  human  affairs  was  not  to  be 
found;  and  yet  the  general  form  which  such  a  law  must  take  was 
plain  to  see.  There  must  be  a  limit  to  the  population  of  a  region, 
great  or  small;  and  the  curve  of  growth  must  sooner  or  later  "turn 
over,"  approach  the  limit,  and  resolve  itself  into  an  S-shaped  curve. 
The  rate  of  growth  (or  annual  increment)  will  depend  (1)  on  the 
population  at  the  time,  and  (2)  on  "the  still  unutiUsed  reserves  of 
population-support  existing"  in  the  available  land.  Here  we  have, 
to  all  intents  and  purposes,  the  growth-factor  and  retardation-factor 
of  Verhulst,  and  they  lead  to  the  same  formula,  or  the  same 
differential  equation,  as  his*. 

A  hundred  years  have  passed  since  Verhulst  dealt  with  the  first 
U.S.  census  returns,  and  found  them  verifying  the  Malthusian 
expectation  of  a  doubhng  every  twenty-five  years.  That  "grande 
vitesse  d'accroissement "  continued  through  five  decennia;  but  it 
ceased  some  seventy  years  ago,  and  a  retarding  influence  has  been 
manifest  through  all  these  seventy  years  (Fig.  29).  It  is  more 
recently,  only  after  the  census  of  1910,  that  the  curve  seemed  to  be 

*  Raymond  Pearl  and  L.  J.  Reed,  on  the  Rate  of  growth  of  the  population  of 
the  U.S.  since  1790,  and  its  mathematical  representation,  Proc.  Nat.  Acad.  Sci. 
VI,  pp.  275-288,  1920;  ibid,  vin,  pp.  365-368,  1922;  Metron,  m,  1923.  In  the 
first  edition  of  Pearl's  Medical  Biometry  and  Statistics,  1923  (2nd  ed.  1930),  Verhulst 
is  not  mentioned.  See  also  his  Studies  in  Human  Biology,  Baltimore,  1924,  Natural 
History  of  Population,  1939,  and  other  works. 


150 


THE  RATE  OF  GROWTH 


[CH. 


1800  1850  1900  1940 

Fig.  29.     Decennial  increments  of  the  population  of  the  United  States. 
*   The  Civil  War.     *  *   The  "slump". 


zuu 

— 1 — 1 

-1 — r- 

1      1      1      1      1 

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- 

1 

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_ 

y 



y 

- 

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/ 

~ 

• 

150 

— 

/                           _ 

/ 

_ 

1 

/ 
/ 

¥            ^ 

- 

t 

/ 

§ 

- 

.  ^ 

/ 

1    '^^ 

/ 

f 

1 

- 

0 

- 

^ 

- 

- 

P 

- 

50 

- 

- 

"""i 

1      1 

1      1      1      1      1 

1  1  1  1  1   1  r  1 

1     1     1 

1800  1850  1900  1950  2000 

Fig.  30.     Conjectural  population  of  the  United  States, 
according  to  the  Verhulst- Pearl  Law. 


Ill]       POPULATION   OF  THE  UNITED   STATES      151 

finding  its  turning-point,  or  point  of  inflection;  and  only  now,  since 
1940,  can  we  say  with  full  confidence  that  it  has  done  so. 

A  hundred  years  ago  the  conditions  were  still  relatively'  simple, 
but  they  are  far  from  simple  now.  Immigration  was  only  beginning 
to  be  an  important  factor;  but  immigrants  made  a  quarter  of  the 
whole  increase  of  the  population  of  the  United  States  during  eighty 
of  these  hundred  years*.  Wars  and  financial  crises  have  made  their 
mark  upon  the  curve;  manners  and  customs,  means  and  standards 
of  living,  have  changed  prodigiously.  But  the  S-shaped  curve 
makes  its  appearance  through  all  of  these,  and  the  Verhulst-Pearl 
formula  meets  the  case  with  surprising  accuracy. 

Population  of  the  United  States 

I  In  ten  years 


Calculated 

No.  of 

A 

Total 

■\ 

Increase 
by  multi- 

by logistic 

immigrants 

increase  of 

plication 

Population 

curve 

landed 

population  Percentage 

in  25 

Year 

xlOOO 

(Udny  Yule) 

xlOOO 

xlOOO 

increase 

years 

1790 

3,929 

3,929 

— 

— 

— 

— 

1800 

5,308 

5,336 

— 

1,379 

351 

— 

1810 

7,240 

7,223 

— 

1,932 

36-4 



1820 

9,638 

9,757 

250 

2,398 

331 

208 

1830 

12,866 

13,109 

228 

3,228 

33-5 

205 

1840 

17,069 

17,506 

538 

4.203 

32-7 

2-02 

1850 

23,192 

23,192 

1,427 

6,123 

35-9 

206 

1860 

31,443 

30,418 

2,748 

8,251 

35-6 

210 

1870 

38,558 

39,372 

2,123 

7,115 

32-6 

1-91 

1880 

50,156 

50,177 

2,741 

11,598 

301 

1-83 

1890 

62.948 

62,769 

5,249 

12,792 

25-5 

1-80 

1900 

75,995 

76,870 

3,694 

13,047 

20-7 

1-71 

1910 

91,972 

91,972 

8,201 

15,977 

210 

1-63 

1920 

105,711 

— 

6,347 

13,739 

14-9 

1-52 

1930 

122,975 

— 

— 

17,264 

161 

1-46 

1940 

131,669 

— 

— 

8,694 

71 

1-33 

A  colony  of  yeast  or  of  bacteria  is  a  population  in  its  simplest 
terms,  and  Verhulst's  law  was  rediscovered  in  the  growth  of  a 
bacterial  colony  some  years  before  Raymond  Pearl  found  it  in  a 
population  of  men,  by  Colonel  M'Kendrick  and  Dr  Kesava  Pai,  who  put 
their  case  very  simply  indeed  f.    The  bacillus  grows  by  geometrical 

*  Without  counting  the  children  born  to  those  immigrants  after  landing,  and 
before  the  next  census  return. 

t  A.  G.  M'Kendrick  and  M.  K.  Pai,  The  rate  of  multiplication  of  micro-organisms : 
a  mathematical  study,  Proc.  R.S.E.  xxxi,  pp.  649-655,  1911.  (The  period  of 
generation  in  B.  coli,  answering  to  Malthus's  twenty-five  years  for  men,  was  found 
to  be  22^  minutes.)  Cf.  also  Myer  Coplans,  Journ.  of  Pathol,  and  Bacleriol.  xiv, 
p.  1,  1910  and  H.  G.  Thornton,  Ana.  of  Applied  Biology,  1922,  p.  265. 


152  THE  RATE  OF  GROWTH  [ch. 

progression  so  long  as  nutriment  is  enough  and  to  spare;  that  is  to 
say,  the  rate  of  growth  is  proportional  to  the  number  present: 

dt        ^ 

But  in  a  test-tube  colony  the  supply  of  nourishment  is  hmited, 
and  the  rate  of  multiphcation  is  bound  to  fall  off.  If  a  be  the 
original  concentration  of  food-stuff,  it  will  have  dwindled  by  time  t 
to  (a  —  y).     The  rate  of  growth  will  now  be 

%  =  by{<^-y), 

which  means  that  the  rate  of  increase  iS  proportional  to  the  number 
of  organisms  present,  and  to  the  concentration  of  the  food-supply. 
It  is  Verhulst's  case  in  a  nutshell;  the  differential  equation  so 
indicated  leads  to  an  S-shaped  curve  which  further  experiment 
confirms;  and  Sach's  "grand  period  of  growth"  is  seen  to  accom- 
phsh  itself*. 

The  growth  of  yeast  is  studied  in  the  everyday  routine  of  a 
brewery.  But  the  brewer  is  concerned  only  with  the  phase  of 
unrestricted  growth,  and  the  rules  of  compound  interest  are  all  he 
needs,  to  find  its  rate  or  test  its  constancy.  A  population  of  1360 
yeast-cells  grew  to  3,550,000  in  35  hours:  it  had  multiplied -2610 
times.    Accordingly, 

1^8^51^^^^  =  0-098  =  log  1-254. 

That  is  to  say,  the  population  had  increased  at  the  rate  of  25-4  per 
cent,  per  hour,  during  the  35  hours. 

The  time  (^2)  required  to  double  the  population  is  easily  found : 

log  2         0-301      ^  ^^  , 

*  The  sigmoid  curve  illustrates  a  theorem  which,  obvious  as  it  may  seem,  is  of  no 
small  philosophical  importance,  to  wit,  that  a  body  starting  from  rest  must,  in  order 
to  attain  a  certain  velocity,  pass  through  all  intermediate  velocities  on  its  way. 
Galileo  discusses  this  theorem,  and  attributes  it  to  Plato:  "Platone  avendo  per 
avventura  avuto  concetto  non  potere  alcun  mobil  passare  daUa  quiete  ad  alcun 

determinate  grado  di  velocita se  non  col  passare  per  tutti  gli  altri  gradi  di 

velocita  minori,  etc.";  Discorsi  e  dimostrazioni,  ed.  1638,  p.  254. 


Ill]  A  POPULATION  OF  YEAST  153 

The  duplication-period  thus  determined  is  known  to  brewers  as 
the  generation-time. 

Much  care  is  taken  to  ensure  the  maximal  growth.  If  the  yeast 
sink  to  the  bottom  of  the  vat  only  its  upper  layers  enjoy  unstinted 
nutriment;  a  potent  retardation-factor  sets  in,  and  the  exponential 
phase  of  the  growth-curve  degenerates  into  a  premature  horizontal 
asymptote.  Moreover,  both  the  yeast  and  the  bacteria  differ  in 
this  respect  from  the  typical  (or  perhaps  only  simpHfied)  case  of 
man,  that  they  not  only  begin  to  suffer  want  as  soon  as  there  comes 
to  be  a  deficiency  of  any  one  essential  constituent  of  their  food*,  but 
they  also  produce  things  which  are  injurious  to  their  own  growth 
and  in  time  fatal  to  their  existence.  Growth  stops  long  before  the 
food-supply  is  exhausted ;  for  it  does  so  as  soon  as  a  certain  balance 
is  reached,  depending  on  the  kind  or  quahty  of  the  yeast,  between 
the  alcohol  and  the  sugar  in  the  cellf. 

If  we  use  the  compound-interest  law  at  all,  we  had  better  think 
of  Nature's  interest  as  being  paid,  not  once  a  year  nor  once  an  hour 
as  our  elementary  treatment  of  the  yeast-population  assumed,  but 
continuously;  and  then  we  learn  (in  elementary  algebra)  that  in 
time  t,  at  rate  r,  a  sum  P  increases  to  Pe'^\  or  P<  ^  ^o^^*. 

Applying  this  to  the  growth  of  our  sample  of  1360  yeast  cells, 
we  have 

PtIPo  =  2610,  log  2610  =  3-417,  which,  multipHed  by  the  modulus 
2-303  =  7-868.     Dividing  by  n  =  35,  the  number  of  hours, 

7-868/35  =  0-225  =  r. 

The  rate,  that  is  to  say,  is  22-5  per  cent,  per  hour,  continuous 
compound  interest.    It  becomes  a  well-defined  physiological  constant, 
and  we  may  call  it,  with  V.  H.  Blackman,  an  index  of  efficiency. 
Our  former  result,  for  interest  at  hourly  intervals,  was  25-4  per 

*  According  to  Liebig's  "law  of  the  minimum." 

t  T.  Carlson,  Geschwindigkeit  und  Grosse'der  Hefevermehrung,  Biochem.  Ztschr. 
Lvn,  pp.  313-334,  1913;  A.  Slator,  Journ.  Chem.  Soc.  cxix,  pp.  128-142,  1906; 
Biochem.  Journ.  vii,  p.  198, 1913;  0.  W.  Richards,  Ann.  of  Botany,  xlh,  pp.  271-283, 
1928;  Alf  Klem,  Hvalradets  Skrifter,  nr.  7,  pp.  55-91,  Oslo,  1933;  Per  Ottestad, 
ibid.  pp.  30-54.  For  optimum  conditions  of  temperature,  nutriment,  pB.,  etc.  see 
Oscar  W.  Richards,  Analysis  of  growth  as  illustrated  by  yeast.  Cold  Spring  Harbour 
Symposia,  ii,  pp.  157-166,  1934. 


154 


THE  RATE  OF  GROWTH 


[CH. 


cent. ;  there  is  no  great  difference  between  such  short  intervals  and 
actual  continuity,  but  there  is  a  deal  of  difference  between  continuous 
payment  and  payment  (say)  once  a  year*.  Certain  sunflowers 
(Helianihus)  were  found  to  grow  as  follows,  in  thirty-seven  days: 


Compound  interest  rate  (%) 


Weight  (gm.) 


Continuous 


Discontinuous 


Giant  sunflower 
Dwarf  sunflower 


Seedling 

0033 
0035 


Plant 

17-33 
14-81 


Per  day   Per  wk.    Per  day      Per  wk. 

170  119  18-5  228% 

16-4  114  17-7  214% 


When  the  yeast  population  is  allowed  to  run  its  course,  it  yields 
a  simple  S-shaped  curve ;   and  the  curve  of  first  differences  derived 


rTi*-2*5<rm-2(r 

m-o" 

m 

m-fcr 

m+2(r   m+2*5(r 

( 

1 
3             1 

2 

3 

4 

5             6             7 

Fig.  31.     The  growth  of  a  yeast-population.     After  Per  Ottestad. 

from  this  is,  necessarily,  a  bell-shaped  curve,  so  closely  resembling 
the  Gaussian  curve  that  any  difference  between  them  becomes  a 
deUcate  matter.  Taking  the  numbers  of  the  population  at  equal 
intervals  of  time  from  asymptotic  start  to  asymptotic  finish,  we 
may  treat  this  series  of  numbers  like  any  other  frequency  distribu- 
tion.   Finding  in  the  usual  way  the  mode  and  standard  deviation, 

*  Cf.  V.  H.  Blackman,  The  compound  interest  law  and  plant  growth,  Ann.  of 
Botany,  xxxiii,  pp.  353-360,  1919.  The  first  papers  on  growth  by  compound 
interest  in  plants  were  by  pupils  of  Noll  in  Bonn:  e.g.  von  Kreusler,  Wachstum  der 
Maispflanze,  Landw.  JB.  1877-79;  P.  Gressler,  Suhstanz-quotienten  von  Helianthus, 
Diss.  Bonn,  1907  etc. 


Ill]  A  POPULATION  OF  FLIES  155 

we  draw  the  corresponding  Gaussian  curve;  and  the  close  "fit" 
between  the  observed  population-curve  and  the  calculated  Gaussian 
curve  is  sufficiently  shewn  by  Mr  Per  Ottestad's  figure  (Fig.  31). 
This  is  a  very  remarkable  thing.  We  began  to  think  of  the  curve 
of  error  as  a  function  with  which  time  had  nothing  to  do,  but  here 
we  have  the  same  curve  (or  to  all  intents  and  purposes  the  same) 
with  time  for  one  of  its  coordinates.  We  might  (I  think)  add  one 
more  to  the  names  of  the  curve  of  error,  and  call  it  the  curve 
of  optimum ;  it  represents  on  either  hand  the  natural  passage  from 
best  to  worst,  from  Ukehest  to  least  likely. 

A  few  flies  (Drosophila)  in  a  bottle  illustrate  the  rise  and  fall  of 
a  population  more  complex  than  yeast,  as  Raymond  Pearl  has 
shewn*  The  colony  dwindles  to  extinction  if  food  be  v/ithheld; 
if  it  be  sufficient,  the  numbers  rise  in  a  smooth  S-shaped  curve; 
if  it  be  plentiful  and  of  the  best,  they  end  by  fluctuating  about  an 
unstable  maximum.  "The  population  waves  up  and  down  about 
an  average  size,"  as  Raymond  Pearl  says,  as  Herbert  Spencer  had 
foreseen t,  and  as  Vito  Volterra's  differential  equations  explain. 
The  growth-rate  slackens  long  before  the  hunger  hne  is  reached; 
crowding  affects  the  birth-rate  as  well  as  the  death-rate,  and  a 
bottleful  of  flies  produces  fewer  and  fewer  offspring  per  pair  the 
more  flies  we  put  into  the  bottle  {.  It  is  true  also  of  mankind,  as 
Dr  WiUiam  Farr  was  the  first  to  shew,  that  overcrowding  diminishes 
the  birth-rate  and  shortens  the  "expectation  of  Hfe§ ."  It  happened 
so  in  the  United  States,  pari  passu  with  the  growth  of  immigration, 
incipient  congestion  acting  (or  so  it  seemed)  as  an  obstacle,  or  a 
deterrent,  to  the  large  families  of  former  days.  Nevertheless,  children 
still  pullulate  in  the  slums.  The  struggle  for  existence  is  no  simple 
affair,  and  things  happen  which  no  mathematics  can  foretell. 

*  Raymond  Pearl  and  S.  L.  Parker,  in  Proc.  Nat.  Acad.  Sci.  vni,  pp.  212-219, 
1922;   Pearl,  Journ.  Exper.  Zool.  Lxm,  pp.  57-84,  1932. 

t  "Wherever  antagonistic  forces  are  in^  action,  there  tends  to  be  alternate 
predominance." 

X  In  certain  insects  an  optimum  density  has  been  observed;  a  certain  amount 
of  crowding  accelerates,  and  a  greater  amount  retards,  the  rate  of  reproduction. 
Cf.  D.  Stewart  Maclagan,  Effect  of  population-density  on  rate  of  reproduction, 
Proc.  R,  S.  (B),  CXI,  p.  437,  1932;  W.  Goetsch,  Ueber  wachstumhemmende  Factoren, 
Zool.  Jakrb.  (Allg.  Zool.),  xlv,  pp.  799-840,  1928. 

§  Dr  W.  Farr,  Fifth  Report  of  the  Registrar-General,  1843,  p.  406  (2nd  ed.). 


156  THE  RATE  OF  GROWTH  [ch. 

An  analogous  S-shaped  curve,  given  by  the  formula  L^  =  kg^, 
was  introduced  by  Benjamin  Gompertz  in  1825* ;  it  is  well  known  to 
actuaries,  and  has  been  used  as  a  curve  of  growth  by  several  writers 
in  preference  to  the  logistic  curve.  It  was  devised,  and  well  devised, 
to  express  a  "law  of  human  mortality",  and  to  signify  the  number 
surviving  at  any  given  age  (x),  "if  the  average  exhaustions  of  a 
man's  power  to  avoid  death  were  such  that  at  the  end  of  infinitely 
small  intervals  of  time  he  lost  equal  portions  (i.e.  equal  proportions) 
of  his  remaining  power 'to  oppose  destruction."  The  principle 
involved  is  very  important.  Death  comes  by  two  roads.  One  is 
by  cha^ce  or  accident,  the  other  by  a  steady  deterioration,  or 
exhaustion,  or  growing  inability  to  withstand  destruction;  and 
exhaustion  comes  (roughly  speaking)  as  by  the  repeated  strokes  of 
an  air-pump,  for  the  life-tables  shew  mortality  increasing  in  geo- 
metrical progression,  at  least  to  a  first  approximation  and  over 
considerable  periods  of  years.  Gompertz  relied  wholly  on  the 
experience  of  "life-contingencies,"  but  the  same  deterioration  of 
bodily  energies  is  plainly  visible  as  growth  itself  slows  down;  for 
we  have  seen  how  growth-rate  in  infancy  is  such  as  is  never  after- 
wards attained,  and  we  may  speak  of  growth-energy  and  its  gradual 
loss  or  decrement,  by  an  easy  but  significant  alteration  of  phrase. 
To  deal  with  the  declining  growth-rate,  as  Gompertz  did  with  the 
falling  expectation  of  life,  and  so  to  measure  the  remaining  energy 
available  from  time  to  time,  would  be  a  greater  thing  than  to  record 
mere  weights  and  sizes;  it  raises  the  problem  from  mere  change  of 
physical  magnitudes  to  an  estimation  of  the  falling  or  fluctuating 
physiological  energies  of  the  bodyf.     We  have  seen  how  in  only 

*  Benjamin  Gompertz,  On  the  nature  of  the  function  expressive  of  the  law  of 
human  mortality,  Phil.  Trans,  xxxvi,  pp.  513-585,  1825.  First  suggested  for  use 
in  growth-problems  by  Sewall  Wright,  Journ.  Amer.  Statist,  Soc.  xxi,  p.  493, 
1926.  See  also  C.  P.  Winsor,  The  Gompertz  curve  as  a  growth  curve,  Proc.  Nat. 
Acad.  Sci.  XVIII,  pp.  1-8,  1932;  cf.  {int.  al.)  G.  R.  Da  vies.  The  growth  curve, 
Journ.  Amer.  Statist.  Soc.  xxii,  pp.  370-374,  1927;  F.  W.  Weymouth  and  S.  H. 
Thompson,  Age  and  growth  of  the  Pacific  cockle,  Bull.  Bureau  Fisheries,  xlvi, 
pp.  63f3-641,  1930-31 ;  also  Weymouth,  McMillen  and  Rich,  in  Journ.  Exp.  Biol. 
vni,  p.  228,  1931. 

t  A  bold  attempt  to  treat  the  question  from  the  physiological  side,  and  on 
Gompertz's  lines,  was  made  only  the  other  day  by  P.  B.  Medawar,  The  growth, 
growth-energy  and  ageing  of  the  chicken's  heart,  Proc.  R.S.  (B),  cxxix,  pp.  332- 
355,  1940.  Cf.  James  Gray,  The  kinetics  of  growth,  Journ.  Exp.  Biol,  vi,  pp.  248- 
274,  1929. 


Ill]  THE   GOMPERTZ   CURVE  157 

few  and  simple  cases  can  a  simple  curve  or  single  formula  be  found 
to  represent  the  growth-rate  of  an  organism;  and  how  our  curves 
mostly  suggest  cycles  of  growth,  each  spurt  or  cycle  enduring  for 
a  time,  and  one  following  another.  Nothing  can  be  more  natural 
from  the  physiological  point  of  view  than  that  energy  should  be 
now  added  and  now  withheld,  whether  with  the  return  of  the 
seasons  or  at  other  stages  on  the  eventful  journey  from  childhood 
to  manhood  and  old  age. 

The  symmetry,  or  lack  of  skewness,  in  the  Verhulst-Pearl  logistic 
curve  is  a  weak  point  rather  than  a  strong;  the  Gompertz  curve 
is  a  skew  curve,  with  its  point  of  inflexion  not  half-way,  but  about 
one- third  of  the  way  between  the  asymptotes.  But  whether  in 
this  or  in  the  logistic  or  any  other  equation  of  growth,  the  precise 
point  of  inflexion  has  no  biological  significance  whatsoever.  What 
we  want,  in  the  first  instance,  is  an  S-shaped  curve  with  a  variable, 
or  modifiable,  degree  of  skewness.  After  all,  the  same  difficulty 
arises  in  all  the  use  we  make  of  the  Gaussian  curve :  which  has  to 
be  eked  out  by  a  whole  family  of  skew  curves,  more  or  less  easily 
derived  from  it.  We  are  far  from  being  confined  to  the  Gaussian 
curve  {sensu  stricto)  in  our  studies  of  biological  probabihty,  or  to  the 
logistic  curve  in  the  study  of  population. 

Yet  another  equation  has  been  proposed  to  the  S-shaped  curve 
of  growth,  by  Gaston  Backman,  a  very  dihgent  student  of  the 
whole  subject.  The  rate  of  growth  is  made  up,  he  says,  of  three 
components:  a  constant  velocity,  an  acceleration  varying  with  the 
time,  and  a  retardation  which  we  may  suppose  to  vary  with  the 
square  of  the  time.  Acceleration  would  then  tend  to  prevail  in  the 
earlier  part  of  the  curve,  and  retardation  in  the  latter,  as  in  fact 
they  do;   and  the  equation  to  the  curve  might  be  written: 

log  H  =  ko  +  k^  log  T-k^  log2  T. 

The  formula  is  an  elastic  one,  and  can  be  made  to  fit  many  an 
S-shaped  curve;   but  again  it  is  empirical. 

The  logistic  curve,  as  defined  by  Verhulst  and  by  Pearl,  has 
doubtless  an  interest  of  its  own  for  the  mathematician,  the  statistician 
and  the  actuary.  But  putting  aside  all  its  mathematical  details  and 
all  arbitrary  assumptions,  the  generalised  S-shaped  curve  is  a  very 
symbol  of  childhood,  maturity  and  age,  of  activity  which  rises  to 


158  THE  RATE  OF  GROWTH  [ch. 

fall  again,  of  growth  which  has  its  sequel  in  decay.  The  growth 
of  a  child  or  of  a  nation;  the  history  of  a  railway*,  or  the  speed 
between  stations  of  a  train;  the  spread  of  an  epidemic]",  or  the 
evolutionary  survival  of  a  favoured  type  J — all  these  things  run 
their  course,  in  its  beginning,  its  middle  and  its  end,  after  the  fashion 
of  the  S-shaped  curve.  That  curve  represents  a  certain  common 
pattern  among  Nature's  "mechanisms,"  and  is  (as  we  have  said 
before),  a  "mecanisme  commun  aux  phenomenes  disparates§." 

At  the  same  time — and  this  is  a  very  interesting  part  of  the  story 
— the  S-shaped  curve  is  no  other  than  what  Galton  called  a  curve  of 
distribution,  that  is  to  say  a  curve  of  integration  or  summation- 
curve,  whose  differential  is  closely  akin  to  the  Gaussian  curve  of 
error. 

Such,  to  a  first  approximation,  is  our  S-shaped  population-curve, 
and  such  are  the  many  phenomena  which,  to  a  first  approximation, 
it  helps  us  to  compare.  But  it  is  only  to  a  first  approximation  that 
we  compare  the  growth  of  a  population  with  that  of  an  organism, 
or  for  that  matter  of  one  organism  or  one  pbpulation  with  another. 
There  are  immense  differences  between  a  simple  and  a  complex 
organism,  between  a  primitive  and  a  civihsed  population.  The 
yeast-plant  gives  a  growth-curve  which  we  can  analyse;  but  we 
must  fain  be  content  with  a  qualitative  description  of  the  growth 
of  a  complex  organism  in  its  complex  world ||. 

There  is  a  simphcity  in  a  colony  of  protozoa  and  a  complexity  in 
a  warm-blooded  animal,  a  uniformity  in  a  primitive  tribe  and  a 
heterogeneity  in  a  modern  state  or  town,  which  affect  all  their 
economies  and  interchanges,  all  the  relations  between  milieu  interne 
and  ex^rne.  and  all  the  coefficients  in  any  but  the  simplest  equations  of 
growth  which  we  can  ever  attempt  to  frame.  Every  growth-problem 
becomes  at  last  a  specific  one,  running  its  own  course  for  its  own 
reasons.     Our  curves  of  growth  are  all  alike — but  no  two  are  ever 


*  Raymond  Pearl,  Amer.  Nat.  lxi,  pp.  289-318,  1927. 

t  Ronald  Ross,  Prevention  of  Malaria  (2nd  ed.),  1911,  p.  679. 

j  J.  B.  S.  Haldane,  Trans.  Camb.  Phil.  Soc.  xxm,  pp.  19^1,  1924. 

§  Cf.  {int.  al.)  J.  R.  Miner's  Note  on  birth-rate  and  density  in  a  logistic  population, 
Human  Biology,  iv,  p.  119,  1932;  and  cf.  Lotka,  ibid,  in,  p.  458,  1931. 

II  Cf.  {int.  al.)  C.  E.  Briggs,  Attempts  to  analyse  growth- curves,  Proc.  E.S.  (B), 
en,  pp.  280-285,  1928. 


Ill]  IN  VARIOUS  ORGANISMS  159 

the  same.  Growth  keeps  caUing  our  attention  to  its  own  com- 
plexity. We  see  it  in  the  rates  of  growth  which  change  with  age 
or  season,  which  vary  from  one  hmb  to  another,;  in  the  influence 
of  peace  and  plenty,  of  war  and  famine ;  not  least  in  those  composite 
populations  whose  own  parts  aid  or  hamper  one  another,  in  any 
form  or  aspect  of  the  struggle  for  existence.  So  we  come  to  the 
differential  equations,  easy  to  frame,  more  difficult  to  solve,  easy  in 
their  first  steps,  hard  and  very  powerful  later  on,  by  which  Lotka 
and  Volterra  have  shewn  how  to  apply  mathematics  to  evolutionary 
biology,  but  which  he  just  outside  the  scope  of  this  book*. 

An  important  element  in  a  population,  and  one  seldom  easy  to 
define,  is  its  age-composition.  It  may  vary  one  way  or  the  other; 
for  the  diminution  of  a  population  may  be  due  to  a  decrease  in  the 
birth-rate,  or  to  an  increasing  mortality  among  the  old.  A  remark- 
able instance  is  that  of  the  food-fishes  of  the  North  Sea.  Their 
birth-rate  is  so  high  that  the  very  young  fishes  remain,  to  all 
appearance,  as  numerous  as  ever;  those  somewhat  older  are  fewer 
than  before,  and  the  old  dwindle  to  a  fraction  of  what  they  were 
wont  to  be. 

The  rate  of  growth  in  other  organisms 

The  rise  and  fall  of  growth-rate,  the  acceleration  followed  by 
retardation  which  finds  expression  in  the  S^shaped  curve,  are  seen 
alike  in  the  growth  of  a  population  and  of  an  individual,  and  in 
most  things  which  have  a  beginning  and  an  end.  But  the  law  of 
large  numbers  smooths  the  population-curve;  the  individual  Hfe 
draws  attention  to  its  own  ups  and  downs;  and  the  characteristic 
sigmoid  curve  is  only  seen  in  the  simpler  organisms,  or  in  parts  or 
"phases"  of  the  more  complex  lives.  We  see  it  at  its  simplest  in 
the  simple  growth-cycle,  or  single  season,  of  an  annual  plant,  which 
cycle  draws  to  its  end  at  flowering;  and  here  not  only  is  the  curve 
simple,  but  its  amplitude  may  sometimes  be  very  large.  The  giant 
Heracleum  and  certain  tall  varieties  of  Indian  corn  grow  to  twelve  feet 

*  See  (int.  al.)  A.  J.  Lotka,  Elements  of  Physical  Biology,  Baltimore,  1925; 
Theorie  analytique  des  associations  biologiques,  Paris,  1934;  Vito  Volterra,  Lemons 
sur  la  theorie  mathematique  de  la  lutte  pour  la  vie,  1931;  Volterra  et  U.  d'Ancona, 
Les  associations  biologiques  au  point  de  vue  mathematique,  1935;  V.  A.  Kostitzin, 
op.  ci7.;\  etc 


160 


THE  RATE  OF  GROWTH 


[CH. 


high  in  a  summer;  the  kudzu  vine  (Pucraria)  may  grow  twelve  inches 
in  twenty-fouE  hours,  and  some  bamboos  are  said  to  have  grown 
twenty  feet  in  three  days  (Figs.  32,  33). 


10 
Days 

Fig.  32.     Growth  of  Lupine.     After  Pfeffer. 


Growth  of  Lupinus  albus.     (From  G.  Backman,  after  Pfeffer) 


Length 

Length 

Day 

(mm.) 

DiflFerence 

Day 

(mm.) 

DiflFerence 

4 

10-5 

— 

14 

132-3 

12-2 

5 

16-3' 

5-8 

15 

140-6 

8-3 

6 

23-3 

7-0 

16 

149-7 

9-1 

7 

32-5 

9-2 

17 

155-6 

5-9 

8 

42-2 

9-7 

18 

158-1 

2-5 

9 

58-7 

14:5 

19 

160-6 

2-5 

10 

77-9 

19-2 

20 

161-4 

0-8 

n 

93-7 

15-8 

21 

161-6 

0-2 

12 

107-4 

13-7 

13 

1201 

12-7 

In  the  pre-natal  growth  of  an  infant  the  S-shaped  curve  is  clearly 
seen  (Fig.  18);  but  immediately  after  birth  another  phase  begins, 
and  a  third  is  imphcit  in  the  spurt  of  growth  which  precedes  puberty. 
In  short,  it  is  a  common  thing  for  one  wave  of  growth  (or  cycle,  as 


Ill]  IN  VARIOUS  ORGANISMS  161 

some  call  it)  to  succeed  another,  whether  at  special  epochs  in  a 
lifetime,  or  as  often  as  winter  gives  place  to  spring*. 
20 


Days 
Fig.  33.     Growth  of  Lupine:  daily  increments. 


45       50 
days 

Fig.  34.     Growth  in  weight  of  a  mouse.     After  W.  Ostwald. 

In  the  accompanying  curve  of  weight  of  the  mouge  (Fig.  34)  we 
see  a  slackening  of  the  rate  of  growth  when  the  mouse  is  about  a 
fortnight  old,  at  which  epoch  it  opens  its  eyes,  and  is  weaned  soon 

*  W.  PfefiFer,  Pflanzenphysiologie,  1881,  Bd.  ii,  p.  78;  A.  Bennett,  On  the  rate 
of  growth  of  the  flower-stalk  of  Vallisneria  spiralis  and  of  Hyacinthus,  Trans.  Linn. 
Soc.  (2),  I,  Botany,  pp.  133,  139,  1880;    cited  by  G.  Backman,  Das  Wachstums- 


162  THE  EATE  OF  GROWTH  [ch. 

after.  At  six  weeks  old  there  is  another  well-marked  retardation; 
it  follows  on  a  rapid  spurt,  and  coincides  with  the  epoch  of  puberty  *. 

In  arthropod  animals  growth  is  apt  to  be  especially  discontinuous, 
for  their  bodies  are  more  or  less  closely  confined  until  released  by 
the  casting  of  the  skin.  The  blowfly  has  its  striking  metamorphoses, 
yet  its  growth  is  wellnigh  continuous;  for  its  larval  skin  is  too  thin 
and  dehcate  to  impede  growth  in  the  usual  arthropod  way.  But 
in  a  thick-skinned  grasshopper  or  hard-shelled  crab  growth  goes  by 
fits  and  starts,  by  steps  and  stairs,  as  Reaumur  was  the  first  to  shew ; 
for,  speaking  of  insects f,  he  says:  "Peut-etre  est-il  vrai  generale- 
ment  que  leur  accroissement,  ou  au  moins  leur  plus  considerable 
accroissement,  ne  se  fait  que  dans  le  temps  qu'ils  muent,  ou  pendant 
im  temps  assez  court  apres  la  mue.  lis  ne  sont  obHges  de  quitter 
leur  enveloppe  que  parce  qu'elle  ne  prend  pas  un  accroissement 
proportionne  a  celui  que  prennent  les  parties  qu'elle  couvre." 
All  the  visible  growth  of  the  lobster  takes  place  once  a  year  at 
moulting-time,  but  he  is  growing  in  weight,  more  or  less,  all  along. 
He  stores  up  material  for  months  together;  then  comes  a  sudden 
rush  of  water  to  the  tissues,  the  carapace  sphts  asunder,  the  lobster 
issues  forth,  devours  his  own  exuviae,  and  lies,  low  for  a  month  while 
his  new  shell  hardens. 

The  silkworm  moults  four 'times,  about  once  a  week,  beginning 
on  the  sixth  or  seventh  day  after  hatching.  There  is  an  arrest  or 
retardation  of  growth  before  each  moult,  but  our  diagram  (i^ig.  35) 
is  too  small  to  shew  the  sHght  ones  which  precede  the  first  and 

problem,  in  Ergebnisse  d.  Physiologie,  xxxiii,  pp.  883-973,  1931.  These  two  cases 
of  Lupinus  and  Vallisneria,  a^e  among  the  many  which  lend  themselves  easily  to 
Backman's  growth -formula,  viz.  Lupinus,  \ogp=  -  2-40  + 1-48  log  T  -  6-61  log^  T  and 
Vallisneria,  log  p  =  +  1-28  +  4-51  log  T  -  2-62  log^  T.  See  for  an  admirable  resume 
of  facts,  Wolfgang  Ostwald,  Ueher  die  zeitliche  Eigenschaften  der  Entwicklungsvorgdnge 
(71  pp.),  1908  (in  Roux's  Vortrdge,  Heft  v);  and  many  later  works. 

*  Cf.  R.  Robertson,  Analysis  of  the  growth  of  the  white  mouse  into  its  con- 
stituent processes,  Journ.  Gen.  Physiology,  vin,  p.  463,  1926.  Also  Gustav 
Backman,  Wachstum  d.  w.  Maus,  Li^nds  Univ.  Arsskrift,  xxxv,  Nr.  12,  1939, 
with  copious  bibliography.  Backman  analyses  the  complicated  growth- curve  of 
the  mouse  into  one  main  and  three  subordinate  cycles,  two  df  which  are  embryonic. 
Cf.  St  Loup,  Vitesse  de  croissance  chez  les  souris.  Bull.  Soc.  Zool.  Fr.  xviii, 
p.  242,  1893;  E.  Le  Breton  and  G.  Schafer,  Trav.  Inst.  Physiol.  Strasburg,  1923; 
E.  C.  MacDowell,  Growth-curve  of  the  suckling  mouse.  Science,  Lxvin,  p.  650, 
1928;  cf.  Journ.  Gen.  Physiol,  xi,  p.  57,  1927;  Ph.  THeritier,  Croissance. .  .dfeins  les 
souris,  Ann.  Physiol,  et  Phys.  Chemie,  v,  p.  i,  1929. 

I  Memoires,  iv,  p.  191. 


i 


Ill] 


OF  CERTAIN  INSECTS 


163 


second.     Before  entering  on  the  pupal  or  chrysalis  stage,  when  the 
worm  is  about  seven  weeks  old,  a  remarkable  process  of  purgation 


e 

mgms 

4000 

1 

3000 

- 

• 

/ 

r 

2000 

- 

^ 

1 

\ 

1000 

- 

- 

IV     / 

I 

II 

III      / 

1     ,  .1 

10 


15 


20 


25 


30 


35 


40 
days 


Fig.  35.     Growth  in  weight  of  silkworm.     From  Ostwald,  after  Luciani 
and  Lo  Monaco. 


takes  place,  with  a  sudden  loss  of  water,  and  of  weight,  which 
becomes  the  most  marked  feature  of  the  curve*    That  the  meta- 

*  Luciani  e  Lo  Monaco,  Arch.  Ital.  de  Biologie,  xxvii,  p.  340,  1897;  see  also 
Z.  Kuwana,  Statistics  of  the  body-weight  of  the  silkworm,  Japan.  Journ.  Zool. 
VII,  pp.  311-346,  1937.  Westwood,  in  1838,  quoted  similar  data  from  Count 
Dandolo:  according  to  whom  100  silkworms  weigh  on  hatching  1  grain;  after 
the  first  four  moults,  15^  94,  270  and  1085  grains;  and  9500  grains  when  full-grown. 


164  THE  RATE  OF  GROWTH  [ch. 

morphoses  of  an  insect  are  but  phases  in  a  process  of  growth  was 
clearly  recognised  by  Swammerdam,  in  the  Bihlid  Naturae*. 

A  stick-insect  (Dexippus)  moults  six  or  seven  times  in  as  many 
months ;  it  lengthens  at  every  ilioult,  and  keeps  of  the  same  length 
until  the  next.  Weight  is  gained  more  evenly;  but  before  each 
moult  the  creature  stops  feeding  for  a  day  or  two,  and  a  little  weight 
is  lost  in  the  casting  pf  the  skin.  After  its  last  moult  the  stick- 
insect  puts  on  more  weight  for  a  while;  but  growth  soon  draws  to 
an  end,  and  the  bodily  energies  turn  towards  reproduction. 

We  have  careful  measurements  of  the  locust  from  moult  to  moult, 
and  know  from  these  the  relative  growth-rates  of  its  parts,  though 
we  cannot  plot  these  dimensions  against  time.  Unlike  the  meta- 
morphosis of  the  silkworm,  the  locust  passes  through  five  larval 
stages  (or  "instars")  all  much  ahke, '  until  in  a  final  moult  the 
"hoppers"  become  winged.  Here  are  three  sets  of  measurements, 
of  Hmbs  and  head,  from  stage  to  stage  f. 

Growth  of  locust,  from  one  moult  to  another 

Length  (mm.)  Percentage-grpwth  Ratios 


Anterior 

Median 

^ 

r 

Anterior 

Median 

"* 

Anterior 

Median 

"* 

Stage 

femur 

femur 

Head 

femur 

femur 

Head 

femur 

femur 

Heac 

I 

1-44 

3-98 

1-44 

— 

— 

•^- 

2-76 

1-00 

II 

2-06 

5-69 

1-94 

1-44 

1-43 

1-35 

2-76 

0-94 

III 

308 

8-22 

2-70 

1-40 

1-44 

1-39 

2-67 

0-88 

IV 

4-53 

11-94 

3-71 

1-47 

1-45 

1'37 

2-76 

0-82 

V 

6-40 

17-22 

4-89 

1-41 

1-44 

1-32 

2-69 

0-76 

Adult 

8-03 

22-85 

5-59 

1-25 

1-33 

V14 

2-84 

0-70 

As  a  matter  of  fact  the  several  parts  tend  to  grow,  for  a  time,  at 
a  steady  rate  of  compound  interest,  which  rate  is  not  identical  for 
head  and  hmbs,  and  tends  in  each  case  to  fall  off  in  the  final  moult, 
when  material  has  to  be  found  for  the  wings.  Some  fifty  years  ago, 
W.  K.  Brooks  found  the  larva  of  a  certain  crab  (Squilla)  increasing 
at  each  moult  by  a  quarter  of  its  own  length ;  and  soon  after 
H.  G.  Dyar  declared  that  caterpillars  grow  hkewise,  from  moult  to 
moult,  by  geometrical  progression  % .     This  tendency  to  a  compoimd- 

*  1737,  pp.  6,  579,  etc. 

t  A.  J.  Duarte,  Growth  of  the  migratory  locust.  Bull.  Ent.  Res.  xxix,  pp.  425-456, 
1938. 

%  W.  K.  Brooks,  Challenger  Report  on  the  Stomatopoda,  1886;  H.  G.  Dyar; 
Number  of  moult^  in  lepidopterous  larvae,  Psyche,  v,  p.  424,  1896. 


Ill]  OF  BROOKS'S  LAW  165 

interest  rate  in  the  growth  and  metamorphosis  of  insects  is  known 
as  Dyar's,  sometimes  as  Brooks's,  law.  According  to  Przibram,  an 
insect  moults  as  soon  (roughly  speaking)  as  cell-division  has  doubled 
the  number  of  cells  throughout  the  larval  body.  That  being  so,  each 
stage  or  instar  should  weigh  twice  as  much  as  the  one  before,  and 
each  linear  dimension  should  increase  by  ^2,  or  1-26  times — a 
Ineasure  identical,  to  all  intents  and  purposes,  with  Brooks's  first 
estimate.  As  sl  first  rough  approximation  the  rule  has  a  certain  value. 
According  to  Duarte's  measurements  the  locust's  total  weight  in- 
creases from  moult  to  moult  by  2-31,  2-16,  242,  2-35,  2-21,  or  a 
mean  increase  of  2-29,  the  cube-root  of  which  is  1-32.  Each  phase 
is  doubled  and  more  than  doubled,  in  passing  to  the  next*,  but 
Przibram's  estimate  is  not  far  departed  from. 

Whatever  truth  Przibram's  law  may  have  in  insects,  or  (as  Fowler 
asserted)  in  the  Ostracods,  it  would  seem  to  have  none  in  the 
Cladocera :  and  this  for  the  sufficient  reason  that  the  shell  (on  which 
the  form  of  the  creature  depends)  goes  on  growing  all  through  post- 
embryonic  life  without  further  division  or  multiphcation  of  its  cells, 
but  only  by  their  individual,  and  therefore  collective,  enlargementf. 

Shells  are  easily  weighed  and  measured  and  their  various  dimen- 
sions have  been  often  studied;  only  in  oysters,  pearl-oysters  and 
the  Hke,  have  they  been  so  kept  under  observation  that  their  actual 
age  is  known.  The  oyster-shell  grows  for  a  few  weeks  in  spring  just 
before  spawning  time,  and  again  in  autumn  when  spawning  is  over ; 
its  growth  is  imperceptible  at  other  times  J. 

*  Cf.  H.  Przibram  and  F.  Megusar,  Wachstummessungen  an  Sphodromantis, 
Arch.  f.  Entw.  Mech.  xxxiv,  pp.  680-741,  1912;  etc.  How  the  discrepancy  is 
accounted  for,  by  Bodenheimer  and  others,  need  not  concern  us  here.  But  cf. 
P.*  P.  Calvert,  On  rates  of  growth  among.  .  .the  Odonata,  Proc.  Amer.  Phil.  Soc. 
Lxviii,  pp.  227-274,  1929,  who  finds  growth  faster  in  nine  cases  out  of  ten  than 
Przibram's  rule  lays  down. 

Millet  asserts,  in  support  of  Przibram's  law,  that  in  spiders  mitotic  cell-division 
is  confined  to  the  epoch  of  the  moult,  and  is  then  manifested  throughout  most  of 
the  tissues  (Bull,  de  Biologie  {SuppL),  viii,  p.  1,  1926).  On  the  other  hand,  the 
rule  is  rejected  by  R.  Gurney,  Rate,  of  growth  in  Copepoda,  Int.  Rev.  Hydrohiol. 
XXI,  pp.  189-27,  1929;  Nobumasa  Kagi,  Growth-curves*  of  insect-larvae,  Mem. 
Coll.  Agric.  Kyoto,  No.  1,  1926;   and  others. 

t  Cf.  W.  Rammer,  Ueber  die  Giiltigkeit  des  Brooksschen  Wachsturasgesetzes 
bei  den  Cladoceren,  Arch.  f.  Entw.  Mech.  cxxi,  pp.  111-127,  1930. 

X  Cf.  J.  H.  Orton,  Rhythmic  periods... in  Ostrea,  Jonrn.  Mar.  Biol.  Assoc. 
XV,  pp.  365^27,  1928;  Nature,  March  2,  1935,  p.  340. 


166 


THE  RATE  OF  GROWTH 


[CH. 


The  window-pane  oyster  in  Ceylon  (Placuna  placenta)  has  been 
kept  under  observation  for  eight  years,  during  which  it  grows  from 
two  inches  long  to  six  (Fig.  36).  The  young  grow  quickly,  and  slow 
down  asymptotically  towards  the  end;  an  S-shaped  beginning  to  the 
growth-curve  has  not  been  seen,  but  would  probably  be  found  in 
the  growth  of  the  first  year.  Changes  of  shape  as  growth  goes  on 
are  hard  to  see  in  this  and  other  shells ;  rather  is  it  characteristic  oi 


Fig.  36. 


12  3  4  5  6  7 

Age  in  years 

Growth  of  the  window-pane  oyster;  short  diameter  of  the  shell. 
From  Pearson's  data. 


them  to  keep  their  shape  from  first  to  last  unchanged.  Nevertheless, 
shght  changes  are  there;  in  the  window-pane  oyster  the  shell  grows 
somewhat  rounder;  in  seven  or  eight  years  the  one  diameter  multi- 
plies (roughly  speaking)  by  eleven,  and  the  other  by  ten*. 

Window-pane  oysters  (Placuna) 


Ratio 

117 
109 
107 
1-06 
105 

The  American  slipper-limpet  has  lately  and  quickly  become  a  pest 
on  English  oyster-beds.     Its  mode  of  growth  is  interesting,  though 

*  Joseph  Pearson,  The  growth-rate. .  .oi Placuna  placenta,  Ceylon  Bulletin,  1928. 


Short 

Long 

diameter 

diameter 

(mm.) 

(mm.) 

150 

17-6 

650 

70-5 

102-5 

109-7 

132-5 

139-9 

167-5 

175-2 

Ill 


OF  THE  GROWTH  OF  SHELLFISH 


167 


the  actual  rate  remains  unknown.  It  grows  a  little  longer  and 
narrower  with  age.  Its  weight-length  coefficient  (of  which  we  shall 
have  more  to  say  presently)  increases  as  time  goes  on,  and  appears 
to  follow  a  wavy  course  which  might  be  accounted  for  if  the 
shell  grew  thinner  and  then  thicker  again,  as  if  ever  so  little  more 
lime  were  secreted  at  one  season  than  another.  The  growth  of  a 
shell,  or  the  deposition  of  its  calcium  carbonate,  is  much  influenced 
by  temperature;  clams  and  oysters  enlarge  their  shells  only  so  long 
as  the  temperature  stands  above  a  certain  specific  minimum,  and 
the  mean  size  of  the  same  hmpet  is  very  different  in  Essex  and  in 
the  United  States*.  Curious  peculiarities  of  growth  have  been 
discovered  in  slipper-hmpets.  Young  limpets  clustered  roimd  an 
old  female  grow  slower  than  others  which  Uve  sohtary  and  apart. 
The  solitary  forms  become  in  turn  male,  hermaphrodite  and  at  last 
female,  but  the  gregarious  or  clustered  forms .  develop  into  males, 
and  so  remain;  development  of  male  characters  and  duration  of 
the  male  phase  depend  on  the  presence  or  absence  of  a  female  in 
the  near  neighbourhood. 


Measurements  of  slipper-limpets 
{From  J.  H.  Eraser's  data,  epitomised) 


No. 

Mean  length 

Breadth 

Ratio 

Weight 

measured 

(mm.) 

(mm.) 

L/B 

(gra-) 

WjD 

3 

15-3 

8-8 

1-74 

0-33 

92 

8 

17-6 

9-8 

1-80 

0-46 

84 

9 

19-4 

10-5 

1-85 

.0-63 

88 

16 

21-5 

11-5 

1-87 

0-77 

77 

18 

23-5 

125 

1-88 

104 

80 

41 

25-5 

13-7 

1-86 

1-37 

85 

91 

27-4 

14-5 

1-89 

1-81 

88 

125 

39-4 

15-4 

1-91 

2-33 

92 

98 

31-4 

16-5 

1-90 

3-22 

104 

70 

33-6 

17-8 

1-89 

3-61 

95 

38 

35-5 

18-6 

1-90 

4-28 

95 

10 

37-3 

19-5 

1-91 

4-95 

95 

1 

321 

19-4 

201 

5-35 

90 

Mean        1-87 


89-3 


*  Cf.  J.  H.  Fraser,  On  the  size  of  Urosalpinx  etc.,  Proc.  Malacol.  Soc.  xix, 
pp.  243-254,  1931.  Much  else  is  known  about  the  growth  of  various  limpets, 
their  seasonal  periodicities,  the  change  of  shape  in  certain  species,  and  other 
matters;  cf.  E.  S.  Russell,  Growth  of  Patella,  P.Z.S.  cxcix,  pp.  235-253;  J.  H. 
Orton,  Journ.  Mar.  BioL  Assoc,  xv,  pp.  277-288,  1929;  Noboru  Abe,  Sci.  Rep. 
Tohoku  Imp.  Univ.  Biol,  vi,  pp.  347-363,  1932,  and  Okuso  Hamai,  ibid,  xii, 
pp.  71-95,  1937. 


168 


THE  RATE  OF  GROWTH 


[CH. 


The  growth  of  the  tadpole  *  is  Hkewise  marked  by  epochs  of 
retardation,  and  finally  by  a  sudden  and  drastic  change  (Fig.  37). 
There  is  a  shght  diminuti9n  in  weight  immediately  after  the  little 
larva  frees  itself  from  what  remains  of  the  egg ;  there  is  a  retardation 


mgms. 

1 

6000 

' 

/ 

') 

5000 
4000 

~  • 

/       ^ 

/    1 

j 

:  1 
■j  j 

3000 

- 

/ 

i 

i 

2000 

- 

1 

/ 

1000 

-  1 

u3 

J 

0  10        20        30        40        50        60         70        80       90 

days 

Fig.  37.     Growth  in  weight  of  tadpole.     From  Ostwald,  after  Schaper. 

of  growth  about  ten  days  later,  when  the  external  gills  disappear;  and 
finally  the  complete  metamorphosis,  with  the  loss  of  the  tail,  the  growth 
of  the  legs  and  the  end  of  branchial  respiration,  brings  about  a  loss 
of  weight  amounting  to  wellnigh  half  the  weight  of  the  full-grown 


*  Cf.   (int.   at.)   Barfurth,    Versiiche  iiber  die   Verwandlung  der  Froschlarven 
Arch.  f.  mikroak.  Anat.  xxix,  1887.* 


Ill] 


OF  LARVAL  EELS 


169 


Fig.  38. 


Development  of  eel :  from  Leptocephalus  larvae  to  young  elver. 
After  Johannes  Schmidt. 


170  THE  RATE  OF  GROWTH  [ch- 

larva.  At  the  root  of  the  matter  hes  the  simple  fact  that  meta- 
morphosis involves  wastage  of  tissue,  increase  of  oxidation,  expendi- 
ture of  energy  and  the  doing  of  work.  While  as  a  general  rule  the 
better  the  animals  be  fed  the  quicker  they  grow  and  the  sooner  they 
metamorphose,  Barfurth  has  pointed  out  the  curious  fact  that  a 
short  spell  of  starvation,  just  before  metamorphosis  is  due,  appears 
to  hasten  the  change. 

The  negative  growth,  or  actual  loss  of  bulk  and  weight  which 
often,  and  perhaps  always,  accompanies  metamorphosis,  is  well 
shewn  in  the  case  of  the  eel  *.  The  contrast  of  size  is  great  between 
the  flattened,  lancet-shaped  Leptocephalus  larva  and  the  httle  black, 
cylindrical,  almost  thread-Uke  elver,  whose  magnitude  is  less  than 
that  of  the  Leptocephalus  in  every  dimension,  even  at  first  in  length 
(Fig.  38),  as  Grassi  was  the  first  to  shew. 

The  lamprey's  case  is  hardly  less  remarkable.  The  larval  or 
Ammocoete  stage  lasts  for  three  years  or  more,  and  metamorphosis, 
though  preceded  by  a  spurt  of  growth,  is  followed  by  an  actual 
decrease  in  size.  The  little  brook  lamprey  neither  feeds  nor  grows 
after  metamorphosis,  but  spawns  a  few  months  later  and  then  dies; 
but  the  big  sea-lampreys  become  semi-parasitic  on  other  fishes,  and 
live  and  grow  to  an  unknown  agef. 

Such  fluctuations  as  these  are  part  and  parcel  of  the  general  flux 
of  physiological  activity,  and  suggest  a  finite  stock  of  energy  to  be 
spent,  now  more  now  less,  on  growth  and  other  modes  of  expenditure. 
The  larger  fluctuations  are  special  interruptions  in  a  process  which 
is  never  continuous,  but  is  perpetually  varied  by  rhythms  of  various 
kinds  and  orders.  Hofmeister  shewed  long  ago,  for  instance,  that 
Spirogyra  grows  by  fits  and  starts,  in  periods  of  activity  and  rest 
alternating  with  one  another  at  intervals  of  so  many  minutes  { 
(Fig.  39).  And  Bose  tells  us  that  plant-growth  proceeds  by  tiny  and 
perfectly  rhythmical  pulsations,  at  intervals  of  a  few  seconds  of  time. 

*  Johannes  Schmidt,  Contributions  to  the  life-history  of  the  eel,  Rapports 
du  Conseil  Intern,  pour  V exploration  de  la  mer,y,  pp.  137-274,  Copenhagen,  1906; 
and  other  papers. 

t  Cf.  {int.  al.)  A.  Meek,  The  lampreys  of  the  Tyne,  Rep.  Dove  Marine  Laboratory 
(N.S.),  VI,  p.  49,  1917;  cf.  L.  Hubbs,  in  Papers  of  the  Michigan  Academy,  iv, 
p.  587,  1924. 

%  Die  Lehre  der  Pflanzenzelle,  1867.  Cf.  W.  J.  Koningsberger,  Tropismus  und 
Wachstum  (Thesis),  Utrecht,  1922. 


Ill] 


OF  PERIODIC  GROWTH 


171 


A  crocus  grows,  he  says,  by  little  jerks,  each  with  an  amphtude  of 
about  0-002  mm.,  every  twenty  seconds  or  so,  each  increment  being 
followed  by  a  partial  recoil*  (Fig.  40).     If  this  be  so  we  have  come 


20   40    60    80   100   120   140   160   180   200   220   240  260 

minutes 

Fig.  39.     Growth  in  length  (mm.)  of  Spirogyra.     From  Ostwald,  after  Hofmeister. 


Fig.  40. 


seconds. 

Pulsations  of  growth  in  Crocus,  in  micro-millimeter? 
After  Bose. 


down,  so  to  speak,  from  a  principle  of  continuity  to  a  principle  of  dis- 
continuity, and  are  face  to  face  with  what  we  might  call,  by  rough 
analogy,  "quanta  of  growth."  We  seem  to  be  in  touch  with  things 
of  another  order  than  the  subject  of  this  bookt. 

*  J.  C.  Bose,  Plant  Response,  1906,  p.  417;  Growth  and  Tropic  Movements  of 
Plants,  1929. 

t  There  is  an  apparent  and  perhaps  a  real  analogy  between  these  periodic 
phenomena  of  growth  and  the  well-known  phenomenon  of  periodic,  or  oscillatory, 
chemical  change,  as  described  by  W.  Ostwald  and  others;  cf.  (e.g.)  Zeitschr.  f. 
phys.  Chem.  xxxv,  pp.  33,  204,  1900. 


172  THE  RATE  OF  GROWTH  [ch. 

We  may  want  now  and  then  to  make  use  of  scanty  data,  and  find 
a  rougli  estimate  better  than  none.  The  giant  tortoises  of  the 
Galapagos  and  the  Seychelles  grow  to  a  great  age,  and  some  have 
weighed  5001b.  and  more;  but  the  scanty  records  of  captive 
tortoises  shew  much  variation,  depending  on  food  and  climate  as 
well  as  age,  Ninety  young  tortoises  brought  from  the  Galapagos 
in  1928  to  the  southern  United  States  weighed  on  the  average 
18J  lb.,  and  grew  to  44-3  lb.  in  two  years.  Six  taken  to  Honolulu 
weighed  26 J  lb.  each  in  1929,  and  63  lb.  each  the  following  year. 
Another,  kept  in  CaHfornia,  weighed  29  lb.  and  360  lb.  seven  years 
later,  but  only  gained  65  lb.  more  in  the  next  seven  years.     Growth, 


1899  1906  1913 

Fjg.  41.     Approximate  growth  in  weight  of  Galapagos  tortoise. 

as  usual,  is  quick  to  begin  with,  slower  lat^r  on,  and  in  the  old  giants 
must  be  slow  indeed.  If  we  plot  (Fig.  41 )  the  three  successive  weights 
of  the  CaHfornian  specimen,  at  first  they  help  us  little;  but  we  can 
fit  an  S-shaped  curve  to  the  three  points  as  a  first  approximation, 
and  it  suggests,  with  some  plausibility,  that,  at  29  lb.  weight  the 
tortoise  was  from  two  to  three  years  old.  A  loggerhead  turtle, 
which  reaches  a  great  size,  was  found  to  grow  from  a  few  grammes 
to  42  lb.  in  three  years,  and  to  double  that  weight  in  another  year 
and  a  half;  these  scanty  data  are  in  fair  accord,  ^o  far  as  they  go, 
with  those  for  the  giant  tortoises*. 

*  For  these  and  other  data,  see  C.  H.  Townsend,  Growth  and  age  in  the  giant 
tortoises  of  the  Galapagos,  Zoologica,  ix,  pp.  459-466,  1931;  G.  H.  Parker,  Growth 
of  the  loggerhead  turtle,  Amer.  Naturalist,  lxvii,  pp.  367-373,  1929;  Stanley  F. 
Flower,  Duration  of  Life  in  Animals,  in,  Reptiles,  P.Z.S.  (A),  1937,  pp.  1-39. 


Ill] 


OF  WHALES  AND  TORTOISES 


173 


The  horny  plates  of  the  tortoise  grow,  to  begin  with,  a  trifle  faster 
than  the  bony  carapace  below,  and  are  consequently  wrinkled  into 
folds.  There  is  some  evidence,  at  least  in  the  young  tortoises,  that 
these  folds  come  once  a  year,  which  is  as  much  as  to  say  that  there 
is  one  season  of  the  year  when  the  growth-rates  of  bony  and  horny 
carapace  are  especially  discrepant.  This  would  give  an  easy  estimate 
of  age;  but  it  is  plainer  in  some  species  than  in  others,  and  it  never 
lasts  for  long. 


re  m  years 
Fig.  42.     Growth-rate  (approximate)  of  blue  and  finner  whales. 

The  blue  whale,  or '  Sibbald's  rorqual,  largest  of  all  animals, 
grows  to  100  ft.  long  or  thereby,  the  females  being  a  httle  bigger 
than  the  males.  The  mother  goes  with  young  eleven  months.  The 
calf  measures  22  to  25  ft.  at  birth,  and  weighs  between  three  and 
four  tons;  it  is  bom  big,  were  it  smaller  it  might  lose  heat  too 
quickly.  It  is  weaned  about  nine  months  later,  and  is  said  to  be 
some  16  metres,  or  say  53  ft.,  long  by  then.  It  is  believed  to  be 
mature  at  two  years  old,  by  which  time  it  is  variously  stated  to  be 
60  or  even  75  ft.  long ;  the  modal  size  of  pregnant  females  is  about 
80  ft.  or  rather  more.  How  long,  the  whale  takes  to  grow  the 
further  15  or  20  feet  which  bring  it  to  its  full  size  is  not  known; 
but,  even  so  far,  the  rapid  growth  and  early  maturity  seem  very 
remarkable  (Fig.  42).     The  Norwegian  whalers  give  us  statistics, 


174 


THE  RATE  OF  GROWTH 


[CH. 


month  by  month  during  the  Antarctic  season,  of  the  sizes  of  pregnant 
females  and  the  foetuses  they  contain;  and  from  these  I  draw  the 
following  averages: 

Antarctic  blue  whales;  length  of  mother  and  of  foetus 
(Season  1938-39) 


Number 
measured 

Mother 

Foetus 

Nov. 

1,  1938 
16 

59 
86 

84-0  ft. 
83-0 

4-2  ft 
4-6 

Dec. 

1 
16 

359 
522 

84-0 
83-6 

61 

7-0 

Jan. 

1,  1939 
16 

403 
317 

83-7 
84-8 

8-4 
9-3 

Feb. 

1 
16 

184 
125 

83-9 
83-9 

11-2' 
12-3 

Mar. 

1 

71 

83-6 

14-5 

2126 


83-8 


^       .- 


NOV 


DEC 


JAN 


FEB 


MAR 


Fig.  43.     Pre-natal  growth  of  blue  whale.     Average  monthly  sizes, 
from  data  in  International  Whaling  Statistics,  xiv,  1940. 

The  observations  are  rough  but  numerous.  At  the  lower  end  of 
the  scale  measurements  are  few,  and  the  value  indicated  is  probably 
too  high;  but  on  the  whole  the  curve  of  growth  taUies  with  other 
estimates,  and  points  to  birth  about  June  or  July,  and  to  conception 
about  the  same  time  last  year  (Fig.  43).  The  mean  size  of  the 
mother- whales  does  not  alter  during  the  five  months  in  question; 


Ill]  THE  GROWTH  OF  WHALES  175 

they  do  not  seem  to  be  increasing,  though  at  84  ft.  they  still  have 
another  10  feet  or  more  to  grow.  They  may  grow  slower,  and  live 
longer,  than  is  often  supposed*. 

On  the  other  hand,  if  we  draw  from  the  same  official  statistics 
the  mean  size  of  mother-whale  and  foetus  at  some  given  epoch  of 
the  year  (e.g.  March  1934),  there  appears  to  be  a  marked  correlation 
between  them,  such  as  would  indicate  very  considerable  growth 
of  the  mother  during  the  months  of  pregnancy.  The  matter  deserves 
further  study,  and  the  data  need  confirmation. 

Blue  whales;  length  of  mother  and  foetus  {March  1934) 


Size  of 

Size  (ft.) 

foetus  (ft.) 

Number 
observed 

X 

smoothed 
in  threes 

Mother 

Foetus 

1 

74 

10 

' — 

1 

75 

7-0 

4-4 

5 

76 

5-2 

6-4 

7 

77 

7-3 

6-4 

9 

78 

6-7 

6-9 

10 

79 

6-7 

6-8 

21 

80 

71 

7-2 

27 

81 

7-7 

7-5 

28 

,    82 

7-6 

7-9 

33 

83 

'8-4 

8-3 

38 

84 

8-9 

8-6 

46 

85 

8-5 

8-6 

37 

86 

8-5 

8-8 

19 

87 

9-5 

9-3 

18 

88 

9-9 

10-3 

12 

89 

11-4 

10-9 

18 

90 

11-5 

111 

9 

91 

10-4 

111 

2 

92 

11-5 

— 

341 

On  the  growth  of  fishes,  and  the  determination  of  their  age 

We  may.  keep  a  child  under  observation,  and  weigh  and  measure 
him  every  day;  but  more  roundabout  ways  are  needed  to  determine 
the  age  and  growth  of  the  fish  in  the  sea.  A  few  fish  may  be  caught 
and  marked,  on  the  chance  of  their  being  caught  again;    or  a  few 

*  The  growth  of  the  finner  whale,  or  common  rorqual,  is  estimated  as  follows 
(Hamburg  Museum) :  at  birth,  6  m.;  at  6  months,  12  m. ;  at  one  and  two  years  old, 
15  and  19  m. ;  when  full-grown,  at  6-8  ( ?)  years  old,  21  m.  For  data,  see  Hvalradets 
Skrifter  and  Jhtemational  Whaling  Statistics,  passim ;  also  N.  Mackintosh  and  others 
in  Discovery  Reports;  also  Sigmund  Rusting,  Statistics  of  whales  and  whale- 
foetuses,  Rapports  du  Conseil  Int.  1928;   etc. 


176  THE  RATE  OF  GROWTH  [ch. 

more  may  be  kept  in  a  tank  or  pond  and  watched  as  they  grow. 
Both  ways  are  slow  and  difficult.  The  advantage  of  large  numbers 
is  not  obtained ;  and  it  is  needed  all  the  more  because  the  rate  of 
growth  turns  out  to  be  very  variable  in  fishes,  as  it  doubtless  is 
in  all  cold-blooded  or  " poecilothermic "  animals:  changing  and 
fluctuating  not  only  with  age  and  season,  but  with  food-supply, 
temperature  and  other  known  and  unknown  conditions.  Trout  in 
a  chalk-stream  so  differ  from  those  in  the  peaty  water  of  a  highland 
burn  that  the  former  may  grow  to  three  pounds  weight  while  the 
latter  only  reach  four  ounces,  at  three  years  old  or  four*. 

It  is  found  (and  easily  verified)  that  shells  on  the  seashore,  kind 
for  kind,  do  not  follow  normal  curves  of  frequency  in  respect  of 
magnitude,  but  fall  into  size-groups  with  intervals  between,  so 
constituting  a  multimodal  curve.  The  reason  is  that  they  are  not 
born  all  the  year  round,  as  we  are,  but  each  at  a  certain  annual 
breeding-season ;  so  that  the  whole  population  consists  of  so  many 
"groups,"  each  one  year  older,  and  bigger  in  proportion,  than 
another.  In  short  we  find  size-groups,  and  recognise  them  as  age- 
groups.  Each  group  has  its  own  spread  or  scatter,  which  increases 
with  ^ize  and  age;  even  from  the  first  one  group  tends  to  overlap 
another,  but  the  older  groups  do  so  more  and  more,  for  they  have 
had  more  time  and  chance  to  vary.  Hence  this  way  of  determining 
age  gets  harder  and  less  certain  as  the  years  go  by;  but  it  is  a  safe 
and  useful  method  for  short-lived  animals,  or  in  the  early  lifetime 
of  the  rest.  Aristotle's  fishermen  used  it  when  they  recognised 
three  sorts  or  sizes  of  tunnies,  the  auxids,  pelamyds  and  full-grown 
fi^h;  and  when  they  found  a  scarcity  of  pelamyds  in  one  year  to 
be  followed  by  a  failure  of  the  tunny-fishery  in  the  nextf. 

Shells  lend  themselves  to  this  method,  as  Louis  Agassiz  found  when 
he  gathered  periwinkles  on  the  New  England  shore.  Winckworth 
found  the  Paphiae  in  Madras  harbour  "of  two  sizes,  one  group  just 
under  15  mm.  in  length,  the  other  nearly  all  over  30  mm.  A  small 
sample,  dredged  ^ve  months  earlier  hora  the  same  ground,  was  inter- 
mediate between  the  other  two."  When  the  mean  sizes  of  the  two 
groups  were  plotted  against  time,  the  lesser  group  being  shifted 

*  Cf.  C.  A.  Wingfield,  Effect  of  environmental  factors  on  the  growth  of  brown 
trout,  Journ.  Exp.  Biol,  xvii,  pp.  435-448,  1939. 
I  Aristotle,  Hist.  Anim.  vi,  571  a. 


Ill 


THE  GROWTH  OF  FISHES  177 


back  a  year,  a  growth-curve  extending  over  two  seasons  was  obtained ; 
when  extrapolated,  it  seemed  to  start  from  zero  about  May  or  June, 
and  this  date,  at  the  beginning  of  the  hot  season,  was  in  all  proba- 
bihty  the  actual  spawning  time.  Growth  stopped  in  winter,  a 
common  thing  in  our  northern  climate  but  surprising  at  Madras, 
where  the  sea-temperature  seldom  falls  below  24°  C.  Shells  over 
40  mm.  long  were  rare,  and  over  50  mm.  hardly  to  be  found — an 
indication  that  Paphia  seldom  lives  over  a  third  season.  Here  then, 
though  the  numbers  studied  were  all  too  few,  the  method  tells  us 
with  httle  doubt  or  ambiguity  the  age  of  a  sample  and  the  growth- 
rate  of  the  species  to  which  it  belongs*. 

Dr  C.  J.  G.  Petersen  of  Copenhagen  brought  this  method  into  use 
for  the  study  of  fishes,  and  up  to  a  certain  point  it  is  safe  and 
trustworthy  though  seldom  easy.  For  one  thing,  it  is  hard  to  get 
a  "random  sample"  of  fish,  for  one  net  catches  the  big  and  another 
the  small.  The  trawl-net  takes  all  the  big,  but  lets  more  and  more 
of  the  small  ones  through.  The  drift-net  catches  herring  by  their 
heads;  if  too  big,  the  head  fails  to  catch  and  the  fish  goes  free,  if 
too  small  the  fish  shps  through;  so  the  net  selects  a  certain  modal 
size  according  to  its  mesh,  and  with  no  great  spread  or  scatter. 
When  we  use  Petersen's  method  and  plot  the  sizes  of  our  catch  of 
fish,  the  younger  age-groups  are  easily  recognised,  even  though  they 
tend  to  overlap;  but  the  older  fish  are  few,  each  size-group  has  a 
wider  spread,  and  soon  the  groups  merge  together  and  the  modal 
cusps  cease  to  be  recognisable.  There  is  no  way,  save  a  rough 
conjectural  one,  of  analysing  the  composite  curve  into  the  several 
groups  of  which  it  is  composed;  in  short,  this  method  works  well 
for  the  younger,  but  fails  for  the  older  fish. 

Fig.  44  is  drawn  from  a  catch  of  some  500  small  cod,  or  codhng, 
caught  one  November  in  the  Firth  of  Forth,  in  a  small-meshed 
experimental  trawl-net.  They  are  too  few  for  the  law  of  large 
numbers  to  take  full  effect;  but  after  smoothing  the  curve,  three 
peaks  are  clearly  seen,  with  some  sign  of  a  fourth,  indicating  about 


*  R.  Winckworth,  Growth  of  Paphia  undulata,  Proc.  Malacolog.  Soc.  xix, 
pp.  171-174,  1931.  Cf.  {int.  al.)  Weymouth,  on  Mactra  stultorum.  Bull.  Calif. 
Fish  Comm.  vii,  1923;  Orton,  on  Cardium,  Journ.  Mar.  Biol.  Assoc,  xiv,  1927, 
on  Ostrea,  and  on  Patella,  ibid,  xv,  1928;  Ikuso  Hamai,  on  Limpets,  Sci.  Rep. 
Tohoku  Imp.  Univ.  (4),  xn,  1937. 


178  THE  RATE  OF  GROWTH  [ch. 

11cm.,  26,  44  and  60  cm.,  as  the  mean  or  modal  sizes  of  four 
successive  broods.  The  dwindhng  heights  of  the  successive  cusps 
are  a  first  approximation  to  a  "curve  of  mortahty,"  shewing  how 
the  young  are  many  and  the  old  are  few.  Again,  plotting  the  several 
sizes  against  time,  we  should  get  our  curve  of  growth  for  four  years, 
or  a  first  rough  approximation  to  it.  Thus  we  learn  from  a  random 
sample,  caught  in  a  single  haul,  the  mean  (or  modal)  sizes  of  a  fish 
at  several  epochs  of  its  fife,  say  at  two,  three  or  even  more  successive 
intervals  of  a  year ;  and  we  learn  (to  a  first  approximation)  its  rate 
of  growth  and  its  actual  age,  for  the  slope  of  the  growth-curve, 
drawing  to  the  base-fine,  points  to  the  time  when  growth  began. 


^,  Cod.  Nov.  1906.  F. of  F 


20  JO  40 

Length,  in  centimetres 
Fig.  44.     A  catch  of  cod,  shewing  a  multimodal  curve  of  frequency. 

Another  haul,  soon  after,  will  add  new  points  to  the  curve,  and 
confirm  our  first  rough  approximation. 

An  experiment  in  the  Moray  Firth,  a  month  or  two  later,  shewed 
the  first  three  annual  groups  in  much  the  same  way;  but  it  also 
shewed  another  group,  of  about  90  cm.  long,  and  others  larger  still. 
At  first  sight  these  did  not  seem  to  fit  on  to  our  four  successive 
year-groups,  of  11,  26,  44  and  60 .cm.;  but  they  did  so  after  all, 
only  with  a  gap  between.  They  were  older  fish,  six  and  seven  years 
old,  which  had  come  back  to  the  Moray  Firth  to  breed  after  spending 
a  couple  of  years  elsewhere. 

It  was  thought  at  first  that  every  such  experiment  should  tally 
with  another,  and  bring  us  to  a  more  and  more  accurate  knowledge 
of  the  growth-rate  of  this  fish  or  that;  but  there  were  continual 
discrepancies,  and  it  was  soon  found  that  the  rate  varied  from  place 


Ill] 


THE  GROAVTH  OF  FISHES 


179 


to  place,  from  month  to  month,  and  from  one  year  to  another. 
The  growth-rate  of  a  fish  varies  far  more  than  does  that  of  a  warm- 
blooded animal.  The  general  character  of  the  curve  remains*, 
save  that  the  fish  continues  to  grow  even  in  extreme  old  age,  but 
it  draws  towards  its  upper  asymptote  with  exceeding  slowness. 


Fig.  45.     Growth  of  cod  (after  Michael  Graham);  and  of 
mullet  (after  C.  D.  Serbetis). 

The  following  estimate  of  the  mean  growth  of  North  Sea  cod  is 
based,  by  Michael  Graham,  on  a  great  mass  of  various  evidence; 
and  beside  it,  for  comparison,  is  an  estimate  for  the  grey  mullet, 
by  C.  D.  Serbetis.  The  shape  of  the  curve  (Fig.  45)  is  enough  to 
indicate  that  at  six  years  old  the  cod  is  still  growing  vigorously  f, 
while  the  grey  mullet  has  all  but  ceased  to  grow.     As  a  matter  of 

*  It  is  essentially  an  S-shaped  curve,  as  usual;  but  the  conditions  of  larval  life 
obscure  the  first  beginnings  of  the  S. 

t  Norwegian  results,  based  largely  on  otoliths,  are  different.  Gunner  Rollefsen 
holds  that  the  spawning  cod,  or  skrei,  do  not  reach  maturity,  for  the  most  part, 
till  10  or  11  years  old,  and  grow  by  no  more  than  1  to  3  cms.  a  year  (Fiskeriskrifter, 
Bergen,  1933). 


180  THE  RATE  OF  GROWTH  [ch. 

fact,  90  cm.  is,  or  was  till  lately,  the  median  size  of  cod*  in  our 
Scottish  trawl-fishery;  one-tenth  are  over  a  metre  long  and  the 
largest  are  in  the  neighbourhood  of  120  cm.,  with  an  occasional 
giant  of  150  cm.  or  even  more.  But  it  has  come  to  pass  that  fish 
of  outstanding  size  are  seen  no  more  save  on  the  virgin  fishing 
grounds;  a  Greenland  halibut,  brought  home  to  Hull  in  1938, 
weighed  four  hundredweight,  was  nearly  two  feet  thick,  and  must 
have  been  of  prodigious  age. 


Age  (years)     

1 

2 

3 

4 

5 

6 

Length  of  cod  (cm.) 

18 

36 

55 

68 

79 

89 

Length  of  grey  mullet 

21 

36 

46 

51 

53 

55 

There  are  other  ways  of  determining,  or  estimating,  a  fish's  age. 
The  Greek  fishermen  shewed  Aristotle  f  how  to  tell  the  age  of  the 
purple  Murex,  up  to  six  years  old,  by  counting  the  whorls  and 
sculptured  ridges  of  the  shell,  and  also  how  to  estimate  the  age  of 
a  scaly  fish  by  the  size  and  hardness  of  its  scales ;  and  Leeuwenhoek 
saw  that  a  carp's  scales  J  bear  concentric  rings,  which  increase  in 
number  as  the  fish  grows  old.  In  these  and  other  cases,  as  in  the 
woody  rings  of  a  tree,  some  part  of  plant  or  animal  carries  a  record 
of  its  own  age;  and  this  record  may.  be  plain  and  certain,  or  may 
too  often  be  dubious  and  equivocal. 

The  scales  of  most  fishes  shew  concentric  rings,  sometimes  (as  in 
the  herring)  of  a  simple  kind,  sometimes  (as  in  the  cod)  in  a  more 
complex  pattern;  and  the  ear-bones,  or  otohths,  shew  opaque 
concentric  zones  in  their  translucent  structure.  The  scales  are 
''read"  with  apparent  ease  in  herring,  haddock,  salmon,  the  otohths 
in  plaice  and  hake;  but  the  whole  matter. is  beset  with  difficulties, 
and  every  result  deserves  to  be  checked  and  scrutinised  §. 

*  As  distinguished  from  "codling." 

t  Hist.  Animalium,  5476,  10;   6076,  30. 

X  The  carp-breeder  is  especially  interested  in  the  age  of  his  fish;  for,  like  the 
brewer  with  his  yeast,  his  profit  depends  on  the  rate  at  which  thej'^  grow. 
Leeuwenhoek 's  and  other  early  observations  were  brought  to  light  by  C.  HoflFbauer, 
Die  Alterbestimmung  der  Karpfen  an  seiner  Schuppen,  Jahresber.  d.  schles. 
Fischerei-Vereins,  Breslau,  1899. 

§  Thus,  for  instance,  Mr  A.  Dannevig  says  (On  the  age  and  growth  of  the  cod, 
Fiakeridirektorets  Skrifter,  1933,  p.  82):  "as  to  the  problem  of  the  determination 
of  the  age  of  the  cod  by  means  of  scales  and  otohths,  all  workers  agree  that  the 
method  is  useful.  But  on  a  number  of  fundamental  points  there  are  just  as  many 
divergences  of  opinion  as  there  are  investigators." 


Ill]         OF  THE  SIZE  AND  AGE  OF  HERRING         181 

In  the  following  table,  we  see  (a)  the  sizes,  and  (6)  the  number 

of  scale-rings,  in  a  sample  of  some  550  herring  from  the  autumn 
fishery  off  the  east  of  Scotland. 

Rings  3  4  5  6  7  8  9  10         11         12      Total    Mean 

cm.  rings 

31         —        —        —  1  1         _  1  _        _  1  4         8-5 

30        —        —        —  7  5  6  4         —        —        —        22         7-3 

29        —        —  5         18         13  6  6  1  1  1  51         70 

28        —  3         29         38         11  3  3  1  —        —         88         5-9 

27  2  13        41         34  5  5  2         _         _        _       102         5-1 

26  7         43         64         29         —        —  1  _        _        _       144         5.0 

25  4         36        41         11         —        —  —        —        —        —        92        4-6 

24  2  17         15  4         —        —  —        —        —        —        38        4-8 

23        —  5        —        —        —        —  —        —        —        —  5 4K) 

Total   15   117   195   142    35    20  17    2     1    2 

Mean   25-6  25-4  26-5  27-4  28-6  28-7  28-8  28-5  —        — 


In  this  sample,  the  sizes  of  the  550  fish  are  grouped  in  a  somewhat 
skew  curve,  about  a  mode  at  26  cm. ;  and  the  numbers  of  scale-rings 
group  themselves  in  like  manner,  but  with  rather  more  skewness, 
about  a  modal  number  of  five  rings.  Either  way  we  look  at  it, 
there  is  only  one  "group"  of  fish;  and  it  is  highly  characteristic 
of  the  herring  that  a  single  sample,  taken  from  a  single  shoal, 
exhibits  a  unimodal  curve.  Accepting  in  principle  the  view  that 
scale-rings  tend  to  synchronise  with  age  in  years,  we  may  draw  this 
first  deduction  that  our  sample  consists  in  part  (if  not  in  whole) 
of  five-year  old  fish,  whose  average  length  is  about  26  cm. ;  and 
this  length,  of  26  cm.  for  5-ringed,  or  5-year-old  herring,  agrees  well 
with  many  other  determinations  from  the  same  region.  We  shall 
be  on  the  safe  side  if  we  deal,  after  this  fashion,  with  the  one 
predominant  group,  or  mode,  in  each  sample  of  fish;  and  Fig.  46 
shews  an  approximate  curve  of  growth  for  our  East  Coast  herring 
drawn  in  this  way. 

But  the  further  assumption  is  commonly  and  all  but  universally 
made  that  each  individual  herring  carries  the  record  of  its  age  on  its 
scale-rings.  If  this  be  so,  then  our  sample  of  550  fish  is  a  com- 
posite population  of  some  ten  separate  broods  or  successive  ages, 
all  mixed  up  in  a  shoal.  And  again,  if  so,  the  5-year-olds  in  the 
said  population  average  26-5  cm.  in  length,  the  3-year-olds  25-6  cm., 
the  10-year-olds  28-5  cm. ;  but  these  values  do  not  fit  into  a  normal 


182 


THE  RATE  OF  GROWTH 


[CH. 


curve  of  growth  by  any  means.  Still  more  obvious  is  it  that  the 
several  year-classes  (if  such  they  be)  do  not  tally  with  the  age- 
composition  of  any  ordinary  population,  nor  agree  with  any  ordinary 
curve  of  mortality.  But  even  if  we  had  ten  separate  year-groups 
represented  here,  which  I  most  gravely  doubt,  all  that  we  know  of 
the  selective  action  of  the  drift-net  forbids  us  to  assume  that  we 
are  deahng  with  a  fair  random  sample  of  the  herring  population; 
so  that,  even  though  the  number  of  rings  did  enable  us  to  distinguish 
the  successive  broods,  we  should  still  have  no  right  to  assume  that 


Fig.  46. 


0  1  2  3  4  5  6 

Years 
Mean  curve  of  growth  of  Scottish  (East  Coast)  herring. 


these  annual  broods  actually  combine  in  the  proportions  shewn, 
to  form  the  composite  population. 

It  is  held  by  many  (in  the  first"  instance  by  Einar  Lea)  that  we 
may  deduce  the  dimensions  of  a  herring  at  each  stage  of  its  past 
life  from  the  corresponding  dimensions  of  the  rings  upon  its  scales. 
Some  such  relation  nmst  obviously  exist,  but  it  is  an  approximation 
of  the  roughest  kind.  For  it  involves  the  assumption  not  only  that 
the  scales  add  ring  to  ring  regularly  year  to  year,  and  that  fish 
and  scale  grow  all  the  while  at  corresponding  rates  or  in  direct 
proportion  to  one  another,  but  also  that  the  scale  grows  by  mere 


Ill]         OF  THE  SIZE  AND  AGE  OF  HERRING        183 

accretion,  each  annual  increment  persisting  without  further  change 
after  it  is  once  laid  down.  This  is  what  happens  in  a  moUuscan 
shell,  which  is  secreted  or  deposited  as  mere  dead  substance  or 
"formed  material";  but  it  is  by  no  means  the  case  in  bone,  and 
we  have  httle  reason  to  expect  it  bf  the  bony  mesoblastic  tissue  of 
a  fish's  scale.  It  is  much  more  likely  (though  we  do  not  know  for 
sure)  that  "osteoblasts"  and  "osteoclasts"  continue  (as  in  bone)  to 
play  their  part  in  the  scale's  growth  and  maintenance,  and  that 
some  sort  of  give  and  take  goes  on.     In  any  case,  it  is  a  matter  of 

Mean  ajyparent  length  of  one-year-old  herring,  as  deduced  by 
scale-reading  from  herring  of  various  ages  or  ''year-classes'^'' 


Year-class  (or  number 
of  rings) 

.> 

3 

4 

.■) 

6 

7 

8 

9 

Estimated    length    at 
1  year  old 

140 

13-2 

12-7 

I2-0 

12- 1 

11-8 

11-9 

11-8 

fact  and  observation  that  the  rings  alter  in  breadth  as  the  fish  goes 
on  growing  f ;  that  the  oldest  or  innermost  rings  grow  steadily 
narrower,  while  the  outermost  hardly  change  or  even  widen  a  little ; 
that-  the  relative  breadths  of  successive  rings  alter  accordingly; 
and  it  follows  that  when  we  try  to  trace  the  growth  of  a  herring 
through  its  lifetime  from  its  scales  when  it  is  old,  the  result  is  more 
or  less  misleading,  and  the  values  for  the  earher  years  are  apt  to 
be  much  too  small.  The  whole  subject  is  very  difficult,  as  we  might 
well  expect  it  to  be;  and,  I  am  only  concerned  to  shew  some 
small  part  of  its  difficulty  J. 

While  careful  observations  on  the  rate  of  growth  of  the  higher 
animals  are  scanty,  they  shew  so  fax  as  they  go  that  the  general 
features  of  the  phenomenon  are  much  the  same.  Whether  the 
animal  be  long-lived,  as  man  or  elephant,  or  short-lived  hke  horse  § 

*  From  T.  Emrys  Watkin,  The  Drift  Herring  of  the  S.E.  of  Ireland,  Rapports  du 
Conseil  pour  V Exploration  de  la  Mer,  lxxxiv,  p.  85,  1933. 

t  Cf.  {int.  al.)  Rosa  M.  Lee,  Methods  of  age  and  growth  determination  in  fishes 
by  means  of  scales,  Fishery  Investigations,  Dept.  of  Agr.  and  Fisheries,  1^20. 

X  The  copious  literature  of  the  subject  is  epitomised,  so  far,  by  Michael  Graham, 
in  Fishery  lyivestigations  (2),  xi,  No.  3,  1928. 

§  There  is  a  famous  passage  in  Lucretius  (v,  883)  where  he  compares  the  course 
of  life,  or  rate  of  growth,  in  the  horse  and  his  boyish  master:  Principio  circum 
tribus  actis  impiger  annis  Floret  equus,  puer  hautquaquam,  etc. 


184  THE  RATE  OF  GROWTH  [ch. 

or  dog,  it  passes  through  the  same  phases  of  growth ;  and,  to  quote 
Dr  Johnson  again,  "whatsoever  is  formed  for  long  duration  arrives 
slowly  to  its  maturity*."  In  all  cases  growth  begins  slowly;  it 
attains  a  maximum  velocity  somewhat  early  in  its  course,  and 
afterwards  slows  down  (subject  to  temporary  accelerations)  towards 
a  point  where  growth  ceases  altogether.  But  in  cold-blooded 
animals,  as  fish  or  tortoises,  the  slowing  down  is  greatly  protracted, 
and  the  size  of  the  creature  would  seem  never  to  reach,  but  only 
to  approach  asymptotically,  to  a  maximal  limit.  This,  after  all, 
is  an  important  difference.  Among  certain  still  lower  animals 
growth  ceases  early  but  Hfe  goes  on,  and  draws  (apparently)  to  no 
predetermined  end.  So  sea-anemones  have  been  kept  in  captivity 
for  sixty  or  even  eighty  years,  have  fed,  flourished  and  borne 
offspring  all  the  while,  but  have  shewn  no  growth  at  all. 

The  rate  of  growth  of  various  parts  or  organs  f 

That  the  several  parts  and  organs  of  the  body,  within  and 
without,  have  their  own  rates  of  growth  can  be  amply  demonstrated 
in  the  case  of  man,  and  illustrated  also,  but  chiefly  in  regard  to 
external  form,  in  other  animals.  There  lies  herein  an  endless 
field  for  the  study  of  correlation  and  of  variability  J. 

In  the  accompanying  table  I  show,  from  some  of  Vierordt's  data, 
the  relative  weights  at  various  ages,  compared  with  the  weight  at 
birth,  of  the  entire  body,  and  of  brain,  heart  and  liver;  also  the 
changing  relation  which  each  of  these  organs  consequently  bears, 
as  time  goes  on,  to  the  weight  of  the  whole  body  (Fig.  47) §. 

*  All  of  which  is  tantamount  to  a  mere  change  of  scale  of  the  time-curve. 

f  This  phenomenon,  of  incrementum  inequale,  as  opposed  to  incrementum  in 
universum,  was  most  carefully  studied  by  Haller:  "Incrementum  inequale  multis 
modis  fit,  ut  aliae  partes  corporis  aliis  celerius  increscant.  Diximus  hepar  minus 
fieri,  majorem  pulmonem,  minimum  thymum,  etc."     (Elem.  viii  (2),  p.  34.) 

X  See  {int.  al.)  A.  Fischel,  Variabilitat  und  Wachsthum  des  embryonalen 
Korpers,  Morphol.  Jahrb.  xxiv,  pp.  369-404,  1896;  Oppel,  Vergleickung  des 
Entwickelungsgrades  der  Organe  zu  verschiedenen  Entwickelungszeiten  hei  Wirhel- 
thieren,  Jena,  1891;  C.  M.  Jackson,  Pre-natal  growth  of  the  human  body  and  the 
relative  growth  of  the  various  organs  and  parts,  Amer.  Journ.  of  Anat.  ix,  1909; 
and  of  the  albino  rat,  ibid,  xv,  1913;  L.  A.  Calkins,  Growth  of  the  human  body  in 
the  foetal  period.  Rep.  Amer.  Assoc.  Anat.  1921.  For  still  more  detailed  measure- 
ments, see  A.  Arnold,  Korperuntersuchungen  an  1656  Leipziger  Studenten,  Ztschr. 
f.  Konstitutionslehre,  xv,  pp.  43-113,  1929. 

§  From  Vierordt's  Anatomische  Tabellen,  pp.  38,  39,  much  abbreviated. 


Ill] 


OF  PARTS  OR  ORGANS 


185 


Weight  of  various  organs,  compared  with  the  total  weight  of  the  human 
body' (male).     (From  Vierordfs  Anatomische  Tahellen) 


Percentage  increase 


Percentage  of  body-wt. 


Wt. 

r 

-A.        ,        ,        . 

^ 

, 

A 

s 

Age 

(kgm.) 

Body 

Brain 

Heart 

Liver 

Brain 

Heart 

Live 

0 

31 

10 

10 

10 

10 

12-3 

0-76 

4-6 

1 

9-0 

2-9 

2-5 

1-8 

2-4 

10-5 

0-46 

3-7 

2 

110 

3-6 

2-7 

2-2 

30 

9-3 

0-47 

3-9 

3 

125 

40 

2-9 

2-8 

3-4 

8-9 

0-52 

3-9 

4 

140 

4-5 

3-5 

31 

4-2 

9-5 

0-53 

4-2 

5 

15-9 

51 

3-3 

3-9 

3-8 

7-9 

0-51 

3-4 

6 

17-8 

5-7 

3-6 

3-6 

4-3 

7-6 

0-48 

3-5 

7 

19-7 

6-4 

3-5 

3-9 

4-9 

6-8 

0-47 

3-5 

8 

21-6 

70 

3-6 

40 

4-6 

6-4 

0-44 

30 

9 

23-5 

7-6 

3-7 

4-6 

50 

61 

0-46 

30 

10 

25-2 

8-1 

3-7 

5-4 

5-9 

5-6 

0-51 

3-3 

11 

27-0 

8-7 

3-6 

6-0 

61 

50 

0-52 

3-2 

12 

290 

9-4 

3-8 

(4.1) 

6-2 

4-9 

(0-34) 

30 

13 

331 

10-7 

3-9 

7-0 

7-3 

45 

0-50 

31 

14 

371 

120 

3-4 

9-2 

8-4 

3-5 

0-58 

3-2 

15 

41-2 

13-3 

3-9 

8-5 

9-2 

3-6 

0-48 

32 

16 

45-9 

14-8 

3-8 

9-8 

9-5 

3-2 

0-51 

30 

17 

49-7 

160 

3-7 

10-6 

10-5 

2-8 

0-51 

30 

18 

53-9 

17-4 

3-7 

10-3 

10-7 

2-6 

0-46 

2-8 

19 

57-6 

18-6 

3-7 

11-4 

11-6 

2-4 

0-51 

2-9 

20 

59-5 

19-2 

3-8 

12-9 

110 

2-4 

0-51 

2-6 

21 

61-2 

19-7 

3-7 

12-5 

11-5 

2-3 

0-49 

2-7 

22 

62-9 

20-3 

3o 

13-2 

11-8 

2-2 

0-50 

2-7 

23 

64-5 

20-8 

3-6 

12-4 

10-8 

2-2 

0-46 

2-4 

24 





3-7 

131 

13-0 

— 

— 

— 

25 

66-2 

21-4 

3-8 

12-7 

12-8 

2-2 

0-46 

2-8 

Fig.  47. 


10 

Age  in  years 
Relative  growth  in  weight  of  brain,  heart  and  body  of  man. 
From  Quetelet's  data  (smoothed  curves). 


186  THE  RATE  OF  GROWTH  [ch. 

We  see  that  neither  brain,  heart  nor  Hver  keeps  pace  by  any 
means  with  the  growing  weight  of  the  whole;  there  must  then 
be  other  parts  of  the  fabric,  probably  the  muscles  and  the  bones, 
which  increase  more  rapidly  than  the  general  average.  Heart  and 
liver  grow  nearly  at  the  same  rate,  the  liver  keeping  a  little  ahead 
to  begin  with,  and  the  heart  making  up  on  it  in  the  end;  by  the 
age  of  twenty-five  both  have  multiplied  their  original  weight  at 
birth  about  thirteen  times,  but  the  body  as  a  whole  has  multiplied 
by  twenty-one.  In  contrast  to  these  the  brain  has  only  multiplied 
its  weight  about  three  and  three-quarter  times,  and  shews  but  little 
increase  since  the  child  was  four  or  five,  and  hardly  any  since  it 
was  eight  years  old.  Man  and  the  gorilla  are  born  with  brains  much 
of  a  size;  but  the  gorilla's  brain  stops  growing  very  soon  indeed, 
while  the  child's  has  four  years  of  steady  increase.  The  child's 
brain  grows  quicker  than  the  gorilla's,  but  the  great  ape's  body 
grows  much  quicker  than  the  child's;  at  four  years  old  the  young 
gorilla  has  reached  about  80  per  cent,  of  his  bodily  stature,  and  the 
child's  brain  has  reached  about  80  per  cent,  of  its  full  size. 

Even  during  foetal  life,  as  well  as  afterwards,  the  relative  weight  of  the 
brain  keeps  on  declining.  It  is  about  18  per  cent,  of  the  body- weight  in  the 
third  month,  16  per  cent,  in  the  fourth,  14  per  cent,  in  the  fifth;  and  the 
ratio  falls  slowly  till  it  comes  to  about  12  per  cent,  at  birth,  say  10  per  cent, 
a  year  afterwards,  and  little  more  than  2  per  cent,  at  twenty*.  Many  statistics 
indicate  a  further  decrease  of  brain-weight,  actual  as  well  as  relative.  The 
fact  has  been  doubted  and  denied ;  but  Raymond  Pearl  has  shewn  evidence 
of  a  slow  decline  continuing  throughout  adult  life  f. 

The  latter  part  of  the  table  shews  the  decreasing  weights  of  the 
organs  compared  with  the  body  as  a  whole:  brain,  which  was 
12  per  cent,  of  the  body- weight  at  birth,  falling  to  2  per  cent,  at 
five-and-twenty ;  heart  from  0-76  to  0-46  per  cent.;  liver  from 
4-6  to  2-78  per  cent.  The  thyroid  gland  (as  we  know  it  in  the  rat) 
grows  for  a  few  weeks,  and  then  diminishes  during  all  the  rest  of 
the  creature's  Hfetime;  even  during  the  brief  period  of  its  own 
growth  it  is  growing  slower  than  the  body  as  a  whole. 

It  is  plain,  then,  that  there  is  no  simple  and  direct  relation,  holding 

*■  Cf.  J.  Ariens  Kappers,  Proc.  K.  Akad.  Wetensch.,  Amsterdam,  xxxix,  No.  7,  1936. 
t  R.  Pearl,  Variation  and  correlation  in  brain-weight,  Biometrika,  iv,  pp.  13-104, 
1905. 


Ill]  OF  PARTS  OR  ORGANS  187 

good  throughout  life,  between  the  size  of  the  body  and  its  organs; 
and  the  ratio  of  magnitude  tends  to  change  not  only  as  the  individual 
grows,  but  also  with  change  of  bodily  size  from  one  individual,  one 
race,  one  species  to  another.  In  giant  and  pigmy  breeds  of  rabbits, 
the  organs  have  by  no  means  the  same  ratio  to  the  body- weight ;  but 
if  we  choose  individuals  of  the  same  weight,  then  the  ratios  tend  to 
be  identical,  irrespective  of  breed*.  The  larger  breeds  of  dogs  are 
for  the  most  part  lighter  and  slenderer  than  the  small,  and  the  organs 
change  their  proportions  with  their  size.  The  spleen  keeps  pace 
with  the  weight  of  the  body ;  but  the  liver,  hke  the  brain,  becomes 
relatively  less.  It  falls  from  about  6  per  cent,  of  the  body-weight 
in  little  dogs  to  rather  over  2  per  cent,  in  a  great  hound  f. 

The  changing  ratio  with  increasing  magnitude  is  especially 
marked  in  the  case  of  the  brain,  which  constitutes  (as  we  have  just 
seen)  an  eighth  of  the  body- weight  at  birth,  and  but  one-fiftieth  at 
twenty-five.  This  faUing  ratio  finds  its  parallel  in  comparative 
anatomy,  in  the  general  law  that  the  larger  the  animal  the  smaller 
(relatively)  is  the  brain  J.  A  falhng  ratio  of  brain- weight  during  life 
is  seen  in  other  animals.  Max  Weber  §  tells  us  that  in  the  lion,  at 
five  weeks,  four  months,  eleven  months  and  lastly  when  full-grown, 
the  brain  represents  the  following  fractions  of  the  weight  of  the  body : 
viz.  i/18,  1/80,  1/184  and  1/546.  And  Kellicott  has  shewn  that  in  the 
dogfish,  while  certain  organs,  e.g.  pancreas  and  rectal  gland,  grow 
pari  passu  with  the  body,  the  brain  grows  in  a  diminishing  ratio, 
to  be  represented  (roughly)  by  a  logarithmic  curve ||. 

In  the  grown  man,  Raymond  Pearl  has  shewn  brain-weight  to 
increase  with  the  stature  of  the  indix'idual  and  to  decrease  with 
his  age,  both  in  a  straight-hne  ratio,  or  linear  regression,  as  the 

*  R.  C.  Robb,  Hereditary  size-limitation  in  the  rabbit,  Journ.  Exp.  Biol,  vi, 
1929. 

t  Cf.  H.  Vorsteher,  Einfiuss  d.  Gesamtgrosse  auf  die  Zusammensetzung  des 
Kbrpers;   Diss.,  Leipzig,  1923. 

X  Oliver  Goldsmith  argues  in  his  Animated  Nature  as  follows,  regarding  the  un- 
likelihood of  dwarfs  or  giants:  "Had  man  been  born  a  dwarf,  he  could  not  have 
been  a  reasonable  creature;  for  to  that  end,  he  must  have  a  jolt  head,  and  then  he 
would  not  have  body  and  blood  enough  to  supply  his  brain  with  spirits;  or  if  he 
had  a  small  head,  proportionable  to  his  body,  there  would  not  be  brain  enough  for 
conducting  life.     But  it  is  still  worse  with  giants,  etc." 

§   Die  Sdugethiere,  p.  117. 

II  Amer.  Journ.  of  Anatomy,  viii,  pp.  319-353,  1908. 


188  THE  RATE  OF  GROWTH  [ch. 

statisticians  call  it.     Thus  the  following  wholly  empirical  equations 
give  the  required  ratios  in  the  case  of  Swedish  males : 

Brain-weight  (gms.)  =  1487-8  —  1-94  x  age,  or 
=  915-06  +  2-86  X  stature. 

In  the  two  sexes,  and  in  different  races,  these  empirical  constants 
will  be  greatly  changed*;  and  Donaldson  has  further  shewn  that 
correlation  between  brain-weight  and  body-weight  is  much  closer 
in  the  rat  than  in  manf. 


Weight  of 

• 

entire 

Weight  of 

animal 

brain 

Ratios 

(gm.) 

(gm.) 

J, 

In 

r 

W 

w 

w:W  y^iv:-^W 

M>»»=Fr 

Marmoset 

335 

12-5 

1:26 

1:20 

w  =  2-30 

Spider  monkey 

1,845 

126 

15 

M 

1-56 

Felis  minuta 

1,234 

23-6 

52 

1-2 

2-25 

F.  domestica 

3,300 

31 

107 

2-4 

2-36 

Leopard 

27,700 

164 

168 

12 

200 

Lion 

119,500 

219 

546 

1-3 

217 

Dik-dik 

4,575 

37 

124 

2-7 

2-30 

Steinbok 

8,600 

49-5 

173 

2-9 

2-32 

Impala 

37,900 

148-5 

255 

2-75 

211 

Wildebeest 

212,200 

443 

479 

2-8 

201 

Zebra 

255,000 

541 

472 

2-7 

1-98 

»> 

297,000 

555 

536 

2-8 

200 

Rhinoceros 

765,000 

655 

1170 

3-6 

209 

Elephant 

3,048,000 

5,430 

560 

20 

1-74 

Whale  (Globiocephalus) 

1,000,000 

2,511 

400 

20 

1-77 

Mean  2-23  206 

Brandt,  a  very  philosophical  anatomist,  argued  some  seventy 
years  ago  that  the  brain,  being  essentially  a  hollow  structure,  a 
surface  rather  than  a  mass,  ought  to  be  equated  with  the  surface 
rather  than  the  mass  of  the  animal.  This  we  may  do  by  taking 
the  square-root  of  the  brain-weight  and  the  cube-root  of  the  body- 
weight;  and  while  the  ratios  so  obtained  do  not  point  to  equality, 
they  do  tend  to  constancy,  especially  if  we  hmit  our  comparison  to 
similar  or  related  animals.  Or  we  may  vary  the  method,  and  ask 
(as  Dubois  has  done)  to  what  power  the  brain-weight  must  be  raised 

*  Biometrika,  iv,  pp.  13-105,  1904. 

t  H.  H.  Donaldson,  A  comparison  of  the  white  rat  with  man,  etc.,  Boas  Memorial 
Volume,  New  York,  1906,  pp.  5-26. 


Ill]  OF  PARTS  OR  ORGANS  189 

to  equal  the  body-weight;  and  here  again  we  find  the  same  tendency 
towards  uniformity*. 

The  converse  to  the  unequal  growth  of  organs  is  found  in  their 
unequal  loss  of  weight  under  starvation.  Chossat  found,  in  a 
well-known  experiment,  that  a  starved  pigeon  had  lost  93  per  cent, 
of  its  fat,  about  70  per  cent,  of  hver  and  spleen,  40  per  cent,  of  its 
muscles,  and  only  2  per  cent,  of  brain  and  nervous  tissues  f.  The 
salmon  spends  many' weeks  in  the  river  before  spawning,  without 
taking  food.  The  muscles  waste  enormously,  but  the  reproductive 
bodies  continue  to  grow. 

As  the  internal  organs  of  the  body  grow  at  different  rates,  so  that 
their  ratios  one  to  another  alter  as  time  goes  on,  so  is  it  with  those 
hnear  dimensions  whose  inconstant  ratios  constitute  the  changing 
form  and  proportions  of  the  body.  In  one  of  Quetelet's  tables 
he  shews  the  span  of  the  outstretched  arms  from  year  to  year,  com- 
pared with  the  vertical  stature.  It  happens  that  height  and  span 
are  so  nearly  co-equal  in  man  that  direct  comparison  means  little ; 
but  the  ratio  of  span  to  height  (Fig.  48)  undergoes  a  significant  and 
remarkable  change.  The  man  grows  faster  in  stretch  of  arms  than 
he  does  in  height,  and  span  which  was  less  at  birth  than  stature  b}'' 
about  I  per  cent,  exceeds  it  by  about  4  per  cent,  at  the  age  of 
twenty.  Quetelet's  data  are  few  for  later  years,  but  it  is  clear 
enough  that  span  goes  on  increasing  in  proportion  to  stature.  How 
far  this  is  due  to  actual  growth  of  the  arms  and  how  far  to  increasing 
breadth  of  the  chest  is  another  story,  and  is  not  yet  ascertained. 

*  Cf.  A.  Brandt,  Sur  le  rapport  du  poids  du  cerveau  a  celui  du  corps  chez 
diflFerents  aniraaux,  Bull,  de  la  Soc.  Imp.  des  naturalistes  de  Moscou,  XL,  p.  525, 
1867;  J.  Baillanger,  De  I'etendu  de  la  surface  du  cerveau,  Ann.  Med.  Psychol. 
XVII,  p.  1,  1853;  Th.  van  Bischoff,  Das  Hirngewicht  des  Menschen,  Bonn,  1880 
(170  pp.),  cf.  Biol.  Centralhl.  i,  pp.  531-541,  1881;  E.  Dubois,  On  the  relation 
between  the  quantity  of  brain  and  the  size  of  the  body,  Proc.  K.  Akad.  Wetensch., 
Amsterdam,  xvi,  1913.  Also,  Th.  Ziehen,^  Maszverhaltnisse  des  Gehirns,  in 
Bardeleben's  Handh.  d.  Anatomie  des  Menschen;  P.  Warneke,  Gehirn  u.  Korper- 
gewichtsbestimmungen  bei  Saugern,  Journ.  f.  Psychol,  u.  Neurol,  xiii,  pp.  355-403, 
1909;  B.  Klatt,  Studien  zum  Domestikationsproblem,  Bibliotheca  genetica,  ii, 
1921;  etc.  The  case  of  the  heart  is  somewhat  analogous ;  see  Parrot,  Zooi.  Ja^r&. 
(System.),  vii,  1894;   Piatt,  in  Biol.  Centralhl.  xxxix,  p.  406,  1919. 

t  C.  Chossat,  Recherches  sur  I'inanition,  Mem.  Acad,  des  Sci.,  Paris,  1843, 
p.  438. 


190 


THE  RATE  OF  GROWTH 


[CH. 

The  growth-rates  of  head  and  body  differ  still  more;    for  the 
height  of  the  head  is  no  more  than  doubled,  but  stature  is  trebled, 


Height  of  the  head  in  man  at  various  ages  * 
(After  Quetelet,  p.  207,  abbreviated) 

Men  Women 


Stature 

Head 

Stature 

Head 

Age 

m. 

ra. 

Ratio 

m. 

m. 

Rat 

Birth 

0-50 

Oil 

4-5 

0-49 

Oil 

4-4 

1  year 

0-70 

015 

4-5 

0-69 

015 

4-5 

2  years 

0-79 

017 

4-6 

0-78 

0-17 

4-5 

3     „ 

0-86 

0-18 

4-7 

0-85 

0-18 

4-7 

5     „ 

0-99 

019 

51 

0-97 

019 

51 

10     „ 

1-27 

0-21 

6-2 

1-25 

0-20 

6-2 

20     „ 

1-51 

0-22 

70 

1-49 

0-21 

7  0 

25     ,. 

1-67 

0-23 

7-3 

1-57 

0-22 

71 

30     „ 

1-69 

0-23 

7-4 

1-58 

0-22 

71 

40     „ 

1-69 

0-23 

7-4 

1-58 

0-22 

71 

Fig.  48. 


Ratio  of  stature  in  man,  to  span  of  outstretched  arms. 
From  Quetelet's  data. 


between  infancy  and  manhood.  Diirer  studied  and  illustrated  this 
remarkable  phenomenon,  and  the  difference  which  accompanies  and 


*  A  smooth  curve,  very  similar  to  this,  is  given  by  Karl  Pearson  for  the  growth 
in  "auricular  height"  of  the  girl's  head,  in  Biometrika,  ni,  p.  141,  1904. 


Ill]  OF  PARTS  OR  ORGANS  191 

results  from  it  in  the  bodily  form  of  the  child  and  the  man  is  easy 
to  see. 

The  following  table  shews  the  relative  sizes  of  certain  parts  and 
organs  of  a  young  trout  during  its  most  rapid  development;  and 
so  illustrates  in  a  simple  way  the  varying  growth-rates  in  different 
parts  of  the  body*.  It  would  not  be  difficult,  from  a  picture  of  the 
little  trout  at  any  one  of  these  stages,  to  draw  its  approximate 
form  at  any  other  by  the  help  of  the  numerical  data  here  set 
forth.  In  like  manner  a  herring's  head  and  tail  grow  longer, 
the  parts  betw^een  grow  relatively  less,  and  the  fins  change  their 
places  a  httle;  the  same  changes  take  place  with  their  specific 
differences  in  related  fishes,  and  herring,  sprat  and  pilchard 
owe  their  specific  characters  to  their  rates  of  growth  or  modes  of 
increment  f. 

Trout  (Salmo  fario) ;  proportionate  growth  of  various  organs 
(From  Jenkinson's  data) 


Days 

Total 

1st 

Ventral 

2nd 

Tail 

Breadth 

old 

length 

Eye 

Head 

dorsal 

fin 

dorsal 

fin 

•  of  tail 

40 

100 

100 

100 

100 

100 

100 

100 

100 

63 

130 

129 

148 

149 

149 

108 

174 

156 

77 

155 

147 

189 

(204) 

(194) 

139 

258 

220 

92 

173 

179 

220 

(193) 

(182) 

155 

308 

272 

106 

195 

193 

243 

173 

165 

173 

337 

288 

Sachs  studied  the  same  phenomenon  in  plants,  after  a  method 
in  use  by  Stephen  Hales  a  hundred  and  fifty  years  before.  On  the 
growing  root  of  a  bean  ten  narrow  zones  were  marked  off,  starting 
from  the  apex,  each  zone  a  millimetre  long.  After  twenty-four 
hours'  growth  (at  a  given  temperature)  the  whole  ten  zones  had 
grown  from  10  to  33  mm.,  but  the  several  zones  had  grown  very 
unequally,  as  shewn  in  the  annexed  table  J  (p.  192): 

*  Cf.  J.  W.  Jenkinson,  Growth,  variability  "and  correlation  in  young  trout,  Bio- 
metrika,  vin,  pp.  444-466.  1912. 

I  Cf.  E.  Ford,  On  the  transition  from  larval  to  adolescent  herring,  Journ.  Mar. 
Biol.  Assoc.  XVI,  p.  723;  xviii,  p.  977,  1930-31.  8o  also  in  larval  eels,  tail  and 
body  grow  at  different  rates,  which  rates  differ  in  different  species;  cf.  Johannes 
Schmidt,  Meddel.  Kommlss.  Havsiindersok.  1916;  L.  Bertin,  Bull.  Zool.  France, 
1926,  p.  327. 

I  From  Sachs's  Textbook  of  Botany,  1882,  p.  820. 


192 


THE  RATE  OF  GROWTH 


[CH. 


Graded  growth  of  bean-root 


Increment 

Increment 

Zone 

mm. 

Zone 

mm. 

Apex 

1-5 

6th 

1-3 

2nd 

5-8 

7th 

0-5 

3rd 

8-2 

8th 

0-3 

4th 

3-5 

9th 

0-2 

5th 

1-6 

10th 

01 

"...  I  marked  in  the  same  manner  as  the  Vine,  young  Honeysuckle  shoots, 
etc. . . . ;  and  I  found  in  them  all  a  gradual  scale  of  unequal  extensions,  those  parts 
extending  most  which  were  tenderest,"  Vegetable  Staticks,  Exp.  cxxiii. 

The  lengths  attained  by  the  successive  zones  He  very  nearly 
on  a  smooth  curve  or  gradient;  for  a  certain  law,  or  principle 
of  continuity,  connects  and  governs  the  growth-rates  along  the 
growing  axis.     This  curve  has  its  family  likeness  to  those  differential 


10 


2  3         4  5         6  7         8 

Zones 
Fig.  49.     Rate  of  growth  of  bean-root,  in  successive  zones 
of  1  mm.  each,  beginning  at  the  tip. 

curves  which  we  have  already  studied,  in  which  rate  of  growth  was 
plotted  against  time,  as  here  it  is  plotted  against  successive  spatial 
intervals  of  a  growing  structure ;  and  its  general  features  are  those 
of  a  curve,  a  skew  curve,  of  error.  Had  the  several  growth-rates 
been  transverse  to  the  axis,  instead  of  being  longitudinal  and 
parallel  to  it,  they  would  have  given  us  a  leaf-shaped  structure, 
of  which  our  curve  would  represent  the  outline  on  either  side;  or 
again,  if  growth  had  been  symmetrical  about  the  axis,  it  might  have 
given  us  a  turnip-shaped  soHd  of  revolution.  There  is  always  an 
easy  passage  from  growth  to  form. 


Ill]  CONCERNING  GRADIENTS  193 

A  like  problem  occurs  when  we  deal  with  rates  of  growth  in 
successive  natural  internodes;  and  we  may  then  pass  from  the 
actual  growth  of  the  internodes  to  the  varying  number  of  leaves 
which  they  successively  produce.  Where  we  have  whorls  of  leaves 
at  each  node,  as  in  Equisetum  or  in  many  water-weeds,  then  the 
problem  is  simphfied;  and  one  such  case  has  been  studied  by 
Ra3rmond  Pearl*.  In  Ceratophyllum  the  mean  number  of  leaves 
increases  with  each  successive  whorl,  but  the  rate  of  increase 
diminishes  from  whorl  to  whorl  as  we  ascend.  On  the  main  stem 
the  rate  of  change  is  very  slow;  but  in  the  small  twigs,  or  tertiary 
branches,  it  becomes  rapid,  as  we  see  from  the  following  abbreviated 
table : 

Number  of  leaves  per  whorl  on  the  tertiary  branches  of 
Ceratophyllum 


Order  of  whorl     ... 

1 

2 

3 

4 

5 

6 

Mean  no.  of  leaves 
Smoothed  no. 

6-55 
6-5 

8-07 
8-0 

1 

9-00 
90 

9-20 
9-5 

9-75 
9-8 

10-00 
100 

Raymond  Pearl  gives  a  logarithmic  formula  to  fit  the  case;  but 
the  main  point  is  that  the  numbers  form  a  graded  series,  and  can 
be  plotted  as  a  simple  curve. 

In  short,  a  large  part  of  the  morphology  of  the  organism  depends 
on  the  fact  that  there  is  not  only  an  average,  or  aggregate,  rate  of 
growth  common  to  the  whole,  but  also  a  gradation  of  rate  from  one 
part  to  another,  tending  towards  a  specific  rate  characteristic  of  each 
part  or  organ.  The  least  change  in  the  ratio,  one  to  another,  of 
these  partial  or  locahsed  rates  of  growth  will  soon  be  manifested 
in  more  and  more  striking  differences  of  form;  and  this  is  as  much 
as  to  say  that  the  time-element,  which  is  imphcit  in  the  idea  oigrowthy 
can  never  (or  very  seldom)  be  wholly  neglected  in  our  consideration 
of  form  I . 

A  flowering  spray  of  Montbretia  or  lily-of-the-valley  exemplifies 
a  growth-gradient,  after  a  simple  fashion  of  its  own.     Along  the 

*  On  variation  and  differentiation  in  Ceratophylly,m,  Carnegie  Inst.  Publications, 
No.  58,  1907;  see  p.  87. 

t  Herein  lies  the  easy  answer  to  a  contention  raised  by  Bergson,  an(J  to  which 
he  ascribes  much  importance,  that  "a  mere  variation  of  size  is  one  thing,  and 
a  change  of  form  is  another."  Thus  he  considers  "a  change  in  the  form  of  leaves" 
to  constitute  "a  profound  morphological  difference"  {Creative  Evolution,  p.  71). 


194  THE  RATE  OF  GROWTH  [ch. 

stalk  the  growth-rate  falls  away ;  the  florets  are  of  descending  age, 
from  flower  tO'  bud:  their  graded  differences  of  age  lead  to  an 
exquisite  gradation. of  size  and  form;  the  time-interval  between 
one  and  another,  or  the  "space-time  relation"  between  them  all, 
gives  a  peculiar  quality — ^we  may  call  it  phase-beauty — to  the 
whole.  A  clump  of  reeds  or  rushes  shews  this  same  phase-beauty, 
and  so  do  the  waves  on  a  cornfield  or  on  the  sea.  A  jet  of  water 
is  not  much,  but  a  fountain  becomes  a  beautiful  thing,  and  the 
play  of  many  fountains  is  an  enchantment  at  Versailles. 

On  the  weight-length  coefficient,  or  ponderal  itidex 

So  much  for  the  visible  changes  of  form  which  accompany 
advancing  age,  and  are  brought  about  by  a  diversity  of  rates  of 
growth  at  successive  points  or  in  different  directions.  But  it  often 
happens  that  an  animal's  change  of  form  may  be  so  gradual  as  to 
pass  unnoticed,  and  even  careful  measurement  of  such  small  changes 
becomes  difficult  and  uncertain.  Sometimes  one  dimension  is  easily 
determined,  but  others  are  hard  to  measure  with  the  same  accuracy. 
The  length  of  a  fish  is  easily  measured ;  but  the  breadth  and  depth 
of  plaice  or  haddock  are  vaguer  and  more  uncertain.  We  may  then 
make  use  of  that  ratio  of  weight  to  length  which  we  spoke  of  in  the 
last  chapter:  viz.  that  W  oc  L^,  or  W  ^  kL^,  or  W/L^  =  k,  where 
k,  the  "ponderal  index,"  is  a  constant  to  be  determined  for  each 
particular  case*. 

We  speak  of  this  /;  as  a  "constant,"  with  a  mean  value  specific 
to  each  species  of  animal  and  dependent  on  the  bodily  proportions 
or  form  of  that  animal;  yet  inasmuch  as  the  animal  is  continually 
apt  to  change  its  bodily  proportions  during  life,  k  also  is  continually 
subject  to  change,  and  is  indeed  a  very  delicate  index  of  such 

*  This  relation,  and  how  important  it  is,  were  clearly  recognised  by  Herbert 
Spencer  in  his  Recent  Discussions  in  Science,  etc.,  1871.     The  formula  has  been 

X  y/w 

often,  and  often  independently,  employed:    first  perhaps  in  the  form  — y-  x  100, 

by  R.  Livi,  L'indice  ponderale,  o  rapporto  tra  la  statura  e  il  peso,  Atti  Soc.  Romamt 
Antropologica,  v,  1897.  Values  of  k  for  man  and  many  animals  are  given  by 
H.  Przibram,  in  Form  und  Formel,  1922.  On  its  use  as  an  index  to  the  condition 
or  habit  of  body  of  an  individual,  see  von  Rhode,  in  Abderhalden's  Arbeitsmethoden, 
IX,  4.  The  constant  k  might  be  called,  more  strictly,  ki,  leaving  kf,  and  k,i  for 
the  similar  constants  to  be  deri-ved  from  the  breadth  and  depth  of  the  fish. 


Ill 


THE  PONDERAL  INDEX 


195 


progressive  changes :  delicate — because  our  measurements  of  length 
are  very  accurate  on  the  whole,  and  weighing  is  a  still  more  dehcate 
method  of  comparison. 

Thus,  in  the  case  of  plaice,  when  we  deal  with  mean  values  for 
large  numbers  and  with  samples  so  far  "homogeneous"  that  they 
are  taken  at  one  place  and  time,  we  find  that  k  is  by  no  means 
constant,  but  varies,  and  varies  in  an  orderly  way,  with  increasing 
size  of  the  fish.  The  phenomenon  is  unexpectedly  complex,  much 
more  so  than  I  was  aware  of  when  I  first  wrote  this  book.     Fig.  50 


120  — 


.2 

o     110 


§    100 

^       U 


90 


'  1 

T — :—t     r  -j— T-  1     I 

1  ' 

_ 

December 

^ 

^ 



— 

— 

~-y 

March 

^ 

^'^"^^■"•..•^ 

— 

, ,  1 

1     1    1    1    1    1     1    1     1 

\  , 

1   1    1    1    1 

1   1 

22 


25 


40 


30  35 

Length  (cm.) 

Fig.  50.     Changes  in  the  weight-length  coefficient  of  plaice  with 
increasing  size;  from  March  and  December  samples. 

shews  the  weight-length  coefficient,  or  ponderal  index,  in  two 
large  samples,  one  taken  in  the  month  of  March,  the  other  in 
December.  In  the  latter  sample  k  increases  steadily  as  the  plaice 
grow  from  about  25  to  40  cm.  long;  weight,  that  is  to  say,  increases 
more  rapidly  than  the  cube  of  the  length,  and  it  follows  that  length 
itself  is  increasing  less  rapidly  than  some  other  linear  dimension. 
In  other  words,  the  plaice  grow  thicker,  or  bulkier,  with  length  and 
age.  The  other  sample,  taken  in  the  month  of  March,  is  curiously 
different;  for  now  k  rises  to  a  maximum  when  the  fish  are  some- 
where about  30  cm.  long,  and  then  decHnes  slowly  with  further 
increase  in  size  of  the  fish;  and  k  itself  is  less  in  March  than  in 
December,  the  discrepancy  being  slight  in  the  small  fish  and  great 
in  the  large.  ,  The  "point  of  inflection"  at  30  cm.  or  thereby  marks 


196  THE  RATE  "OF  GROWTH  [ch. 

an  epoch  in  the  fish's  life;  it  is  about  the  size  when  sexual  maturity 
begins,  or  at  least  near  enough  to  suggest  a  connection  between  the 
two  phenomena*. 

A  step  towards  further  investigation  would  be  to  determine  k  for  the  two 
sexes  separately,  and  to  see  whether  or  no  the  point  of  inflection  occurs,  as 
maturity  is  known  to  be  reached,  at  a  smaller  size  in  the  male.  This  d'Ancona 
has  done,  not  for  the  plaice  but  for  the  shad  {Alosa  finta).  He  finds  that  the 
males  are  the  first  to  reach  maturity,  first  to  shew  a  retardation  of  the  rate 
of  growth,  first  to  reach  a  maximal  value  of  th^  ponderal  index,  and  in  all 
probability  the  first  to  diet • 

Again  we  may  enquire  whether,  or  how,  k  varies  with  the  time 
of  year;  and  this  torrelation  leads  to  a  striking  result  J.  For  the 
ponderal  index  fluctuates  periodically  with  the  seasons,  falling 
steeply  to  a  minimum  in  March  or  April,  and  rising  slowly  to  an 
annual  maximum  in  December  (Fig.  51)  §.  The  main  and  obvious 
explanation  hes  in  the  process  of  spawning,  the  rapid  loss  of  weight 
thereby,  and  the  slow  subsequent  rebuilding*  of  the  reproductive 
tissues;  whence  it  follows  that,  without  ever  seeing  the  fish  spawn, 
and  without  ever  dissecting  one  to  see  the  state  of  its  reproductive 
system,  we  may  by  this  statistical  method  ascertain  its  spawning 
season,  and  determine  the  beginning  and  end  thereof  with  con- 
siderable accuracy.  But  all  the  while  a  similar  fluctuation,  of 
much  less  amphtude,  is  to  be  found  in  voung  plaice  before  the 
spawning  age;  whence  we  learn  that  the*  fluctuation  is  not  only 
due  to  shedding  and  replacement  of  spawn,  but  in  part  also  to 
seasonal  changes  in  appetite  and  general  condition. 

Returning  to  our  former  instance,  we  now  see  that  the  March 
and  December  samples  of  plaice,  which  shewed  such  discrepant 
variations  of  the  ponderal  index  with  increasing  size,  happen  to 

♦  The  carp  shews  still  more  striking  changes  than  does  the  plaice  in  the  weight - 
length  coefl&cient:  in  other  words,  still  greater  changes  in  bodily  shape  with 
advancing  age  and  increasing  size;  cf.  P.  H.  Stnithers,  The  Champlain  Watershed, 
Albany,  New  York,  1930. 

■j-  U.  d'Ancona,  II  problema  dell'  accrescimento  dei  pesci,  etc.,  Mem.  R.  Acad, 
dei  Lincei  (6),  n,  pp.  497-540,  1928. 

J  Cf.  Lammel,  Ueber  periodische  Variationen  in  Organismen,  Biol.  Centralbl. 
xxn,  pp.  368-376,  1903. 

§  When  we  restrict  ourselves,  for  simphcity'a  sake,  to  fish  of  one  particular 
size,  we  need  not  determine  the  values  of  k,  for  changes  in  weight  are  obvious 
enough;  but  when  we  have  small  numbers  and  various  sizes  to  deal  with,  the 
determination  of  k  helps  very  much. 


Ill 


THE  PONDERAL  INDEX 


197 


coincide  with  the  beginning  and  end  of  the  spawning  season;  the 
fish  were  full  of  spawn  in  December,  but  spent  and  lean  in  March. 
The  weight-length  ratio  was,  of  necessity,  higher  at  the  former 
season;  and  the  faUing-off  in  condition,  and  in  bulk,  which  the 
March  sample  indicates,  is  more  and  more  pronounced  in  the  larger 
and  therefore  more  heavily  spawn-laden  fish. 


JFMAMJJASONDJ 
Fig.  51.     Periodic  annual  change  in  the  weight-length  ratio  of  plaice. 

Periodi-c  relation  of  weight  to  length  in  plaice  of  55  cm.  long 


Average  weight 

WJL^ 

W/L^  (smoothed) 

decigrams 

Jan. 

204 

1-23 

116 

Feb. 

174 

104 

108 

March 

162 

0-97 

0-99 

April 

159 

0-95 

0-97 

May 

162 

0-98 

0-98 

June 

171 

103 

101 

July 

169 

101 

104 

August 

178 

1-07 

104 

Sept. 

173 

104 

Ml 

Oct. 

203 

1-22 

116 

Nov. 

203 

1-22 

1-21 

Dec. 

200 

1-20 

1-22 

Mean 


180 


1-08 


198  THE  RATE  OF  GROWTH  [ch. 

Plaice  caught  in  a  certain  area,  March  1907  and  December  1905. 
Variation  of  k,  the  weight-length  coefficient,  with  size 


March  sample 

December  sample 

A 

r 

Do. 

Do. 

cm. 

gm. 

WjL^      smoothed 

gm. 

W/L^ 

smoothed 

23 

113 

0-93 









24 

128 

0-93 

0-94 

—  ' 

— 



25 

152 

0-97 

0-96 







26 

178 

0-96 

0-98 

177 

101 



27 

193 

0-98 

0-99 

209 

106 

1-06 

28 

221 

101 

1-00 

241 

MO 

1-08 

29 

250 

102 

101 

264 

108 

109 

30 

271 

1-00 

101 

294 

109 

109 

31 

300 

101 

100 

325 

109 

110 

32 

328 

100 

100 

366 

112 

112 

33 

354 

0-99 

0-99 

410 

114 

113 

34 

384 

0-98 

0-98 

449 

M4 

115 

35 

419 

0-98 

0-98 

501 

117 

117 

36 

454 

0-97 

0-97 

556 

119 

117 

37 

492 

0-95 

0-96 

589 

116 

118 

38 

529 

0-96 

0-96 

652 

119 

119 

39 

564 

0-95 

0-95 

719 

1-21 

1-22 

40 

614 

096 

0-95 

809 

1-26 

— 

41 

647 

0-94 

0-94 

— 





42 

679 

0-92 

0-93 

— 

— 

— 

43 

732 

0-92 

0-93 

— 

•    — 

— 

H 

800 

0-94 

0-94 

— 

— 

— 

4l5 

875 

0-96 

— 

— 

— 

— 

These  weights  and  measurements  of  plaice  are  taken  from  the  Department  of 
Agriculture  and  Fisheries'  Plaice-Report,  i,  pp.  65,  107,  1908;   ii,  p.  92,  1909. 

Japanese  goldfish*  are  exposed  to  a  much  wider  range  of  tem- 
perature than  our  plaice  are  called  on  to  endure ;  they  hibernate  in 
winter  and  feed  greedily  in  the  heat  of  summer.  Their  weight  is 
low  in  winter  but  rises  in  early  spring,  it  falls  as  low  as  ever  at  the 
height  of  the  spawning  season  in  the  month  of  May;  so  for  one 
weight-length  fluctuation  which  the  plaice  has,  the  goldfish  has  a 
twofold  cycle  in  the  year.  The  index  reaches  its  second  and  higher 
maximum  in  August,  and  falls  thereafter  till  the  end  of  the  year. 
That  it  should  begin  to  fall  so  soon,  and  fall  so  quickly,  merely  means 
that  late  autumn  is  a  time  of  groipth ;  the  fish  are  not  losing  weight, 
but  growing  longer  f. 

*  Cf.  Kichiro  Sasaki,  Tohoku  Soi.  Reports  (4),  i,  pp.  239-260,  1926. 

t  Much  has  been  written  on  the  weight-length  index  in  fishes.  See  (int.  al.) 
A.  Meek,  The  growth  of  flatfish,  Northumberland  Sea  Fisheries  Ctee,  1905,  p.  58; 
W,  J.  Crozier,  Correlations  of  weight,  length,  etc,  in  the  weakfish,  Cynoscion 


hi]  the  PONDERAL  index  199 

It  is  the  rule  in  fishes  and  other  cold-blooded  vertebrates  that 
growth  is  asymptotic  and  size  indeterminate,  while  in  the  warm- 
blooded growth  comes,  sooner  or  later,  to  an  end.  But  the 
characteristic  form  is  established  earlier  in  the  former  case,  and 
changes  less,  save  for  the  minor  fluctuations  we  have  spoken  of. 
In  the  higher  animals,  such  as  ourselves,  the  whole  course  of  life 
is  attended  by  constant  alteration  and  modification  of  form;    and 


Fig.  52.     The  ponderal  index,  or  weight-length  coefficient,  in 
man.     From  Quetelet's  data. 

we  may  use  our  weight-length  formula,  or  ponderal  index,  to  illus- 
trate (for  instance)  the  changing  relation  between  height  and  weight 
in  boyhood,  of  which  we  spoke  before  (Fig.  52). 


regalis,  Bull.  U.S.  Bureau  of  Fisheries,  xxxni,  pp.  141-147,  1913;  Selig  Hecht, 
Form  and  growth  in  fishes,  Journ.  of  Morphology,  xxvii,  pp.  379-400,  1916; 
J.  Johnstone  (Plaice),  Trans.  Liverpool  Biolog.  Soc.  xxv,  pp.  186-224,  1911; 
J.  J.  Tesch  (Eel),  Journ.  du  Conseil,  iii,  1927;  Frances  N.  Clark  (Sardine),  Calif. 
Fish.  Bulletin,  No.  19,  1928  (with  full  bibliography).  For  a  discussion  on  statistical 
lines,  apart  from  any  assumptions  such  as  the  "law  of  the  cubes,"  see  G.  Buncker, 
Korrelation  zwischen  Lange  u.  Gewicht,  etc.,  Wissensch.  Meerestmtersuch.  Helgoland, 
XV,  pp.  1-26,  1923. 


THE  RATE  OF  GROWTH 


[CH. 


The  weight-length  coefficient,  or  ponderal  index,  k,  in  young  Belgians 
(From  Quetelefs  figures) 


(years) 

WJL^ 

Age  (years) 

WjL^ 

0 

2-55 

10 

1-25 

1 

2-92 

11 

118 

2 

2-34 

12 

1-23 

3 

2-08 

13 

1-29 

4 

1-87 

14 

1-23 

5 

1-72 

15 

1-23 

6 

1-56 

16 

1-28 

7 

1-48 

20 

1-30 

8 

1-39 

25 

1-36 

9 

1-29 

The  infant  is  plump  and  chubby/  and  the  ponderal  index  is  at 
its  highest  at  a  year  old.  As  the  boy  grows,  it  is  in  stature  that  he 
does  so  most  of  all;  his  ponderal  index  falls  continually,  till  the 
growing  years  are  over,  and  the  lad  "fills  out"  and  grows  stouter 
again.  During  prenatal  Ufe  the  index  varied  httle,  and  less  than 
we  might  suppose : 


Relation  between  length  and  weight  of  the  human  foetus 
(From  Scammon's  data) 


Length 

Weight 

cm. 

gm. 

7-7 

13 

12-3 

41 

17-3 

115 

223 

239 

27-2 

405 

32-3 

750 

37-2 

1163 

42-2 

1758 

46-9 

2389 

51-7 

3205 

2-9 
2-2 
2-2 
2-2 

io 

2-2 
2-3 
2-3 
2-3 
2-3 


As  a  further  illustration  of  the  rate  of  growth,  and  of  unequal 
growth  in  various  directions,  we  have  figures  for  the  ox,  extending 
over  the  first  three  years  of  the  animal's  life,  and  giving  (1)  the 
weight  of  the  animal,  month  by  month,  (2)  the  length  of  the  back, 
from  occiput  to  tail,  and  (3)  the  height  to  the  withers.  To  these 
I  have  added  (4)  the  ratio  of  length  to  height,  (5)  the  weight-length 
coefficient,  k,  and  (6)  a  similar  coefficient,  or  index-number,  k' ,  for 


Ill] 


THE  PONDERAL  INDEX 


201 


the  height,  of  the  animal.  All  these  ratios  change  as  time  goes  on. 
The  ratio  of  length  to  height  increases,  at  first  considerably,  for  the 
legs  seem  disproportionately  long  at  birth  in  the  ox,  as  in  other 

Relations  between  the  weight  and  certain  linear  dimensions  of  the  ox 


{Data  from  i 

Jo^nevin*y 

abbreviated) 

Length 

Age 

Weight 

of  back 

Height 

lonths 

kgm. 

m. 

m. 

L/H 

k  =  WjL^ 

k'  =  Wli 

0 

37 

0-78 

0-70 

Ml 

0-78 

1-08 

1 

55 

0-94 

0-77 

1-22 

0-66 

1-21 

2 

86 

1-09 

0-85 

1-28 

0-67 

1-41 

3 

121 

1-21 

0-94 

1-28 

0-69 

1-46 

4 

150 

1-31 

0-95 

1-38 

0-66 

1-75 

5 

179 

1-40 

104 

1-35 

0-65 

1-60 

6 

210 

1-48 

109 

1-36 

0-64 

1-64 

7 

247 

1-52 

112 

1-36 

0-70 

1-75 

8 

267 

1-58 

115 

1-38 

0-68 

1-79 

9 

283 

1-62 

116 

1-39 

0-66 

1-80 

10 

304 

1-65 

119 

1-39 

0-68 

1-79 

11 

328 

1-69 

1-22 

1-39 

0-67 

1-79 

12 

351 

1-74 

1-24 

1-40 

0-67 

1-85 

13 

375 

1-77 

1-25 

1-41 

0-68 

1-90 

14 

391 

1-79 

1-26 

1-41 

0-69 

1-94 

15 

406 

1-80 

1-27 

1-42 

0-69 

1-98 

16 

418 

1-81 

1-28 

1-42 

0-70 

2-09 

17 

424 

1-83 

1-29 

1-42 

0-69 

1-97 

18 

424 

1-86 

1-30 

1-43 

0-66 

1-94 

19 

428 

1-88 

1-31 

1-44 

0-65 

1-92 

20 

438 

1-88 

1-31 

1-44 

0-66 

1-94 

21 

448 

1-89 

1-32 

1-43 

0-66 

1-94 

22 

464 

1-90 

1-33 

1-43 

0-68 

1-96 

23 

481 

1-91 

1-35 

1-42 

0-69 

1-98 

24 

501 

1-91 

1-35 

1-42 

.0-71 

203 

25 

521 

1-92 

1-36 

1-41 

0-74 

2-08 

26 

534 

1-92 

1-36 

1-41 

0-75 

212 

27 

547 

1-93 

1-36 

1-41 

0-76 

2-16 

28 

555 

1-93 

1-36 

1-41 

0-77 

219 

29 

562 

1-93 

1-36 

1-41 

0-78 

2-22 

30 

586 

1-95 

1-38 

1-41 

0-79 

2-22 

31 

611 

1-97 

1-40 

1-40 

0-80 

2-21 

32 

626 

1-98 

1-42 

1-40 

0-80 

219 

33 

641 

200 

1-44 

1-39 

0-81 

216 

34 

656 

201 

1-45 

1-38 

0-81 

213 

ungulate  animals;  but  this  ratio  reaches  its  maximum  and  falls  off 
a  little  during  the  third  year :  so  indicating  that  the  beast  is  growing 
more   in   height   than  length,   at   a   time   when   growth    in    both 

*  Ch.  Cornevin,  ^^tudes  sur  la  croissanee,  Arch,  de  Physiol,  norm,  et  pathol.  (5),  iv, 
p.  477,  1892.  Cf.  also  R.  Gartner,  Ueber  das  W'achstum  d.  Tiere,  Landxvirtsch. 
Jahresher.  lvii,  p.  707,  1922. 


202 


THE  RATE  OF  GROWTH 


[CH. 


dimensions  is  nearly  over*.  The  ratio  W/H^  increases  steadily, 
and  at  three  years  old  is  double  what  it  was  at  birth.  It  is  the 
most  variable  of  the'  three  ratios;  and  it  so  illustrates  the  some- 
what obvious  but  not  unimportant  fact  that  k  varies  most  for  the 
dimension  which  varies  least,  or  grows  most  uniformly;  in  other 
words,  that  the  values  of  k,  as  determined  at  successive  epochs  for 
any  one  dimension,  are  a  measure  of  the  variability  of  the  other  two. 
The  same  ponderal  index  serves  as  an  index  of  "build,"  or 
bodily  proportion;  and  its  mean  values  have  been  determined  for 
various  ages  and  f6r  many  races  of  mankind.     Within  one  and  the 


ie>u 

1     • 

r- 

• 

y 

y- 

150 

y 

y 

X« 

140 

my 

X' 

y 

130 

^ 

1 

1       1 

1    , 

1 

1         1 

\ 

60 


70 


65 

Height  (inches) 

Fig.  53.     Ratio  of  height  to  weight  in  man.     From  Goringe's  data. 

same  race  it  varies  with  stature ;  for  tall  men,  and  boys  too,  are  apt 
to  be  slender  and  lean,  and  short  ones  to  be  thickset  and  strong. 
And  so  much  does  the  weight-length  ratio  change  with  build  or 
stature  that,  in  the  following  table  of  mean  heights  and  weights  of 
men  between  five  and  six  feet  high,  it  will  be  seen  that  weight, 
instead  of  varying  as  the  cube  of  the  height,  is  (within  the  hmits 
shewn)  in  nearly  simple  linear  relation  to  it  (Fig.  53)  f. 

*  As  a  matter  of  fact,  the  data  shew  that  the  animal  grows  under  7  per  cent, 
in  length,  but  over  11  per  cent,  in  height,  between  the  twentieth  and  the  thirtieth 
month  of  its  age. 

t  Had  the  weights  varied  as  the  cube  of  the  height,  the  tallest  men  should 
have  weighed  close  on  200  lb.,  instead  of  160  lb. 


Ill]  OF  WHALES  AND  ELEPHANTS  203 

Ratio  of  height  to  weight  in  man  * 


No.  of 

Height 

Weight 

instances 

in. 

lb. 

WIH 

Wjm 

59 

60-5 

125 

207 

5-62 

118 

61-5 

129 

213 

5-55- 

220 

62-5 

133 

213 

•     5-45 

285 

63-5 

136 

214 

5-30 

327 

64-5 

139 

215 

519 

386 

65-5 

143 

218 

5-09 

346 

66-5 

146 

2-20 

4-97 

289 

67-5 

153 

2-27 

4-96 

220 

68-5 

151 

2-20 

4-71 

116 

69-5 

156 

224 

4-64 

58 

70-5 

160 

2-27 

4-57 

The  same  index  may  be  used  as  a  measure  of  the  condition,  even 
of  the  quaUty,  of  an  animal;  three  Burmese  elephants  had  the 
following  heights,  weights,  and  reputations!: 


Height 

Weight 

WjH^ 

A 
B 
C 

7  ft.  lOi  in. 

8  1 
7       5 

7,511  lb. 

7,216 

4,756 

1-54 
1-36 
115 

A  famous  elephant 

A  good  elephant 

A  weak,  poor  elephant 

But  a  great  African  elephant,  10  ft.  10  in.  high,  weighed  14,640  Ib.J : 
whence  the  weight-height  coefficient  was  no  more  than  1-15.  That  is 
to  say,  the  African  elephant  is  considerably  taller  than  the  Indian, 
and  the  weight-height  ratio  is  correspondingly  less. 

Lastly,  by  means  of  the  same  index  we  may  judge,  to  a  first  rough 
approximation,  the  weight  of  a  large  animal  such  as  a  whale,  where 
weighing  is  out  of  the  question.  Sigurd  Rusting  has  given  us 
many  measurements,  and  many  foetal  weights,  from  the  Antarctic 
whale-fishery:  among  which,  choosing  at  random,  we  find  that  a 
certain  foetus  of  the  blue  whale,  or  Sibbald's  rorqual,  measured 
4  ft.  6  in.  long,  and  weighed  23  kilos,  or  say  46  lb.  A  whale  of  the 
same  kind,  45  ft.  long,  should  then  weigh  46  x  10^  lb.,  or  about 
23  tons;  and  one  of  90  ft.,  23  x  2^  tons,  or  over  180  tons.  Again 
in  seven  young  unborn  whales,  measuring  from  39  to  54  inches  and 
weighing  from  10  to  23  kilos,  the  mean  value  of  the  index  was  found 

*  Data  from  Sir  C.  Goringe,  The  English  Convict,  H.M.  Stationery  Office,  1913. 
See  also  J.  A,  Harris  and  others.  The,  Measurement  of  Man,  Minnesota,  1930,  p.  41. 
t  Data  from  A.  J.  Milroy,  On  the  management  of  elephants,  Shillong,  1921. 
%  D.  P.  Quireng,  in  Qrcrwth,  iii,  p.  9,  1939. 


204  THE  RATE  OF  GROWTH  [ch. 

to  be  15-2,  in  gramme-inches.     From  this  we  calculate  the  weight 
of  the  great  rorqual,  as  follows : 

1 5*2  X  SOO^ 

At  25  ft.,  or  300  inches,  W  = -— —  =  4,100,000  g. 

lUu 

=  4,100  kg. 

=  4  tons,  nearly. 

At    50  ft.,  TF  =  4  X  23  tons  =  32  tons. 

100  ft.        =  32  X  23  tons  =  256  tons. 

106  tons  (the  largest  known)  W  =  305  tons,  nearly. 

The  two  independent  estimates  are  in  close  agreement. 

Of  surface  and  volume 

While  the  weight-length  relation  is  of  especial  importance,  and 
is  wellnigh  fundamental  to  the  understanding  of  growth  and  form 
and  magnitude,  the  corresponding  relation  of  surface-area  to  weight 
or  volume  has  in  certain  cases  an  interest  of  its  own.  At  the  surface 
of  an  animal  heat  is  lost,  evaporation  takes  place,  and  oxygen  may 
be  taken  in,  all  in  due  proportion  as  near  as  may  be  to  the  bulk 
of  the  animal;  and  again  the  bird's  wing  is  a  surface,  the  area  of 
which  must  be  in  due  proportion  to  the  size  of  the  bird.  In  hollow 
organs,  such  as  heart  or  stomach,  area  is  the  important  thing  rather 
than  weight  or  mass;  and  we  have  seen  how  the  brain,  an  organ 
not  obviously  but  essentially  and  developmentally  hollow,  tends  to 
shew  its  due  proportions  when  reckoned  as  a  surface  in  comparison 
with  the  creature's  mass. 

Surface  cannot  keep  pace  with  increasing  volume  in  bodies  of 
similar  form;  wing-area  does  not  and  cannot  long  keep  pace  with 
the  bird's  increasing  bulk  and  weight,  and  this  is  enough  of  itself 
to  set  limits  to  the  size  of  the  flying  bird.  It  is  the  ratio  between 
square-root-of-surface  and  cube-root-of-volume  which  should,  in 
theory,  remain  constant;  but  as  a  matter  of  fact  this  ratio  varies 
(up  to  a  certain  extent)  with  the  circumstances,  and  in  the  case  of 
the  bird's  wing  with  varying  modes  and  capabilities  of  _flight.  The 
owl,  with  his  silent,  effortless  flight,  capable  of  short  swift  spurts 
of  attack,  has  the  largest  spread  of  wings  of  all ;  the  kite  outstrips 
the  other  hawks  in  spread  of  wing,  in  soaring,  and  perhaps  in  speed. 


Ill]  OF^SURFACE  AND  VOLUME  205 

Stork  and  seagull  have  a  great  expanse  of  wing;  but  other  skilled 
and  speedy  fliers  have  long  narrow  wings  rather  than  large  ones. 
The  peregrine  has  less  wing-area  than  the  goshawk  or  the  kestrel; 
the  swift  and  the  swallow  have  less  than  the  lark. 

Mean  ratio,  VS/^^^W,  between  wing-area  and  weight  of  birds 
{From  Mouillard's  data) 

Ratio 


Owls 

1  1 

species 

2-2 

Hawks 

7 

Gulls 

1 

Waders 

3 

Petrels 

2 

Plovers 

3 

Passeres 

4 

1-3 

Ducks 

2 

1-2 

To  measure  the  length  of  an  animal  is  easy,  to  weigh  it  is  easier  still,  but 
to  estimate  its  surface-area  is  another  thing.  Hence  we  know  but  little  of 
the  surface-weight  ratios  of  animals,  and  what  we  know  is  apt  to  be  uncertain 
and  discrepant.  Nevertheless,  such  data  as  we  possess  average  down  to  mean 
values  which  are  more  uniform  than  we  might  expect*. 

Mean  ratio,  VS/VW,  in  various  animals  [cm.  gm.  units) 


Ape 

11-8 

I             Sheep  (shorn) 

8 

Man 

11 

1             Snake 

12-5 

Dog 

10-11 

1             Frog 

10-6 

Cat,  horse 

10 

Birds 

10 

Rabbit 

9-75 

1             Tortoise 

10 

Cow,  pig,  rat 

9 

A  further  note  on  unequal  growth,  or  heterogony 

An  organism  is  so  complex  a  thing,  and  growth  so  complex  a 
phenomenon,  that  for  growth  to  be  so  uniform  and  constant  in  all 
the  parts  as  to  keep  the  whole  shape  unchanged  would  indeed 
be  an  unlikely  and  an  unusual  circumstance.  Rates  ''^ary,  propor- 
tions change,  and  the  whole  configuration  alters  accordingly.  In  so 
humble  a  creature  as  a  medusoid,.  manubrium  and  disc  grow  at 
different  rates,  and  certain  sectors  of  the  disc  faster  than  others, 
as  when  the  little  Ephyra-lfnya.  "develops"  into  the  great  Aurdia- 
jellyfish.     Many  fishes  grow  from  youth  to  age  with  no  visible, 

*  From  Fr.  G.  Benedict,  Oberflachenbestimmung  verschiedener  Tiergattungen, 
Ergebnisae  d.  Pkysiologiey  xxxvi,  pp.  300-346,  1934  (with  copious  bibliography). 


206  THE  RATE  OF  GROWTH  [ch. 

hardly  a  measurable,  change  of  form*;  but  the  shapes  and  looks 
of  man  and  woman  go  on  changing  long  after  the  growing  age  is 
over,  even  all  their  lives  long.  A  centipede  has  its  many  pairs  of 
legs  alike,  to  all  intents  and  purposes;  they  begin  alike  and  grow 
uniformly.  But  a  lobster  has  his  great  claws  a,nd  his  small,  his 
lesser  legs,  his  swimmerets  and  the  broad  flaps  of  his  tail ;  all  these 
begin  ahke,  and  diverse  rates  of  growth  make  up  the  difference 
between  them.  Moreover,  we  may  sometimes  watch  a  single  Umb 
growing  to  an  unusual  size,  perhaps  in  one  sex  and  not  in  the  other, 
perhaps  on  one  side  and  not  on  the  other  side  of  the  body:  such 
are  the  "horns,"  or  mandibles,  of  the  stag-beetle,  only  conspicuous 
in  the  male,  and  the  great  unsymmetrical  claws  of  the  lobster,  or 
of  that  extreme  case  the  httle  fiddler-crab  {Uca  pugnax).  For  such 
well-marked  cases  of  differential  growth-ratio  between  one  part  and 
another,  JuHan  Huxley  has  introduced  the  term  heterogonyf. 

Of  the  fiddler-crabs  some  four  hundred  males  were  weighed,  in 
twenty-five  graded  samples  all  nearly  of  a  size,  and  the  weights 
of  the  great  claw  and  of  the  rest  of  the  body  recorded  separately. 
To  begin  with  the  great  claw  was  about  8  per  cent.,  and  at  the  end 
about  38  per  cent.,  of  the  total  weight  of  the  unmutilated  body. 
In  the  female  the  claw  weighs  about  8  per  cent,  of  the  whole  from 
beginning  to  end;  and  this  contrast  marks  the  disproportionate, 
or  heterogonic,  rate  of  growth  in  the  male.  We  know  nothing 
about  the  actual  rate  of  growth  of  either  body  or  claw,  we  cannot 
plot  either  against  time;  but  we  know  the  relative  proportions,  or 
relative  rates  of  growth  of  the  two  parts  of  the  animal,  and  this  is 
all  that  matters  meanwhile.  In  Fig.  54,  we  have  set  off  the  successive 
weights  of  the  body  as  abscissae,  up  to  700  mgm.,  or  about  one-third 
of  its  weight  in  the  adult  animal;  and  the  ordinates  represent  the 
corresponding  weights  of  the  claw.  We  see  that  the  ratio  between 
the  two  magnitudes  follows  a  curve,  apparently  an  exponential 
curve;   it  does  in  fact  (as  Huxley  has  shewn)  follow  a  compound 

*  Cf,  S.  Hecht,  Form  and  growth  in  fishes,  Journ.  Morphology,  xxvii,  pp.  379- 
400,  1916;  F.  S.  and  D.  W.  Haramett,  Proportional  length-growth  of  garfish 
(Lepidosteus),  Growth,  in,  pp.  197-209,  1939. 

t  See  Problems  of  Relative  Growth,  1932,  and  many  papers  quoted  therein. 
The  term,  as  Huxley,  tells  us,  had  been  used  by  Pezard;  but  it  Jiad  been  used,  in 
another  sense,  by  Rolleston  long  before  to  mean  an  alternation  of  generations, 
or  production  of  offspring  dissimilar  to  the  parent. 


^73  )([M,j^  J 


III]  OF  UNEQUAL  GROWTH  207 

interest  law,  which  (calhng  y  and  x  the  weights  of  the  claw  and  of 
the  rest  of  the  body)  may  be  expressed  by  the  usual  formula  for 
compound  interest, 

y  =  bx^,  or  log  y  =  \ogb  +  k  log  x; 

and  the  coefficients  (6  and  Jc)  work  out  in  the  case  of  the  fiddler-crab, 
to  begin  with,  at 

2/  =  0-0073  xi-«2.      . 

300 


100        200        300        400        500       600        700 
Weight  of  body  (mgm.) 

Eig.  54.     Relative  weights  of  body  and  claw  in  the  fiddler- 
crab  { Uca  'pugnax). 

But  after  a  certain  age,  or  certain  size,  th^e  coefficients  no  longer 
hold,  and  new  coefficients  have  to  be  found.  Whether  or  no,  the 
formula  is  mathematical  rather  than  biological;  there  is  a  lack  of 
either  biological  or  physical  significance  in  a  growth-rate  which 
happens  to  stand,  during  part  of  an  animaFs  fife,  at  62  per  cent, 
compound  interest. 

Julian  Huxley  holds,  and  many  hoki  with  him,  that  the  exponential 
or  logarithmic  formula,  or  the  compound-interest  .law,  is  of  general 
application  to  cases  of  differential  growth-rates.  I  do  not  find  it  to 
be  so :  any  more  than  we  have  found  organ,  organism  or  population 
to  increase  by  compound  interest  or  geometrical  progression,  save 


208  THE  feATE  OF  GROWTH  [ch. 

under  exceptional  circumstances  and  in  transient  phase.  Undoubtedly 
many  of  Huxley's  instances  shew  increase  by  compound  interest, 
during  a  phase  of  rapid  and  unstinted  growth;  but  I  find  many 
others  following  a  simple-interest  rather  than  a  compound-interest 
law. 

Relative  weights  of  claw  and  body  in  fiddler-crabs  (Uca  pugnax). 
(Data  abbreviated  from  Huxley,  Problems  of  Relative  Growth, 
p.  12) 


rt.  of  body 

less  claw 

Wt.  of 

Ratio 

Wt.  of 

Wt.  of 

Ratio 

(mgm.) 

claw 

% 

body 

claw 

% 

58 

5 

8-6 

618 

243 

39-3 

80 

9 

11-2 

743 

319 

42-9 

109 

14 

12-8 

872 

418 

47-9 

156 

25 

160 

983 

461 

46-9 

200 

38 

190 

1080 

537 

49-7 

238 

53 

22-3 

1166 

594 

50-9 

270 

59 

21-9 

1212 

617 

50-9 

300 

78 

260 

1299 

670 

51-6 

355 

105 

29-7 

1363 

699 

51-3 

420 

135 

321 

1449 

773 

53-7 

470 

165 

351 

1808 

1009 

55-8 

536 

196 

36-6 

2233 

1380 

61-7 

In  the  common  stag-beetle  {Lv^anus  cervus)  we  have  the  following 
measurements  of  mandible  and  elytron  or  wing-case:  which  two 
organs  make  up  the  bulk  of,  and  may  for  our  purpose  be  held 
as  constituting,  the  "total  length"  of  the  beetle.  Here  a  simple 
equation  meets  the  case;  in  other  words,  the  length  of  elytron  or 
of  mandible  plotted  against  total  length  gives  what  is  to  all  intents 
and  purposes  a  straight  hne,  indicating  a  simple-interest  rather 
than  a  compound-interest  rate  of  increase. 

Measurements  of  iS  stag-beetles  (Lucanus  cervus)*  (mm.) 


Number  of  specimens 

1 

4 

5 

10 

5 

7 

11 

5 

Length,  total  (x) 

310 

38-7 

40-5 

42-6 

450 

46-9 

49-2 

53-6 

Length  of  elytron  (y) 

250 

30-9 

31-5 

32-6 

33-8 

351 

36-4 

39-2 

(     „      calculated)  (y') 

26-9 

30-8 

31-7 

32-8 

340 

350 

36-2 

38-5 

Length  of  mandible  (z) 

60 

7-8 

90 

100 

11-2 

11-9 

12-8 

14-4 

(     „     calculated)  (z') 

5-9 

7-7 

9-3 

101 

110 

11-7 

12-6 

14-2 

*  Data,  from  Julian  Huxley,  after  W.  Bateson  and  H.  H.  Brindley,  in  P.Z.S. 
1892,  pp.  585-594. 


Ill]   OF  UNEQUAL  GROWTH  OR  HETEROGONY  209 

From  the  observed  data  we  may  solve,  by  the  method  of  least 
squares,  the  simple  equations 

y  =  a  -{-  bx,     z=  c  +  dx, 

or  in  other  words,  find  the  equations  of  the  straight  lines  in  closest 
agreement  with  the  observed  data.     The  solutions  are  as  follows*: 

y  -  11-02  +  0-512^,   and  z^-  5-64  +  0-368x, 

the  two  coefficients  0-368  and  0-512  signifying  the  difference  between 
the  rates  *of  increase  of  the  two  organs.  The  number  of  samples  is 
not  very  large,  and  some  deviation  is  to  be  expected;  nevertheless, 
the  calculated  straight  lines  come  close  to  the  observed  values. 

40 


10 


20 


Fig. 


30  40  50 

Total  length  (mm.) 

55.     Relative  growth  of  body  and  mandible  in  reindeer-beetle 
{Cydommatus  tarandus). 


The  reindeer-beetle  {Cydommatus  tarandus),  belonging  to  the 
same  family,  shews  much  the  same  thing.  The  mandible  grows  in 
approximately  linear  ratio  to  the  body,  save  that  it  tends  to  be  at 
first  a  little  above,  and  later  on  a  little  below,  this  Unear  ratio 
(Fig.  55). 

Measurements  o/ Cydommatus  tarandus f  (mm.) 


Length  of  mandible  (y) 

3-9 

10-7 

^    141 

19-9 

24-0 

30-7 

34-5 

Total  length  (x) 

20-4 

331 

38-4 

47-3 

54-2 

66-1 

74-0 

Total  length  calculated: 
x  =  l7y  +  U-l 

20-3 

31-9 

37-7 

47-5 

54*5 

65-9 

72-4 

*  As  determined  for  me  by  Dr  A.  C.  Aitken,  F.R.S. 

t  Data,  much  abbreviated,  from  Huxley,  after  E.  Dudieh,  Archivf.  Naturgesch.  (A), 
1923. 


22-0 

48-3 

58-0 

73-5 

891 

102-0 

1120 

420 

65-3 

74-5 

85-5 

99-3 

112-6 

1200 

421 

65-2 

73-7 

87-4 

971 

112-5 

121-2 

210  THE  RATE  OF  GROWTH  [ch. 

The  facial  and  cranial  parts  of  a  dog's  skull  tend  to  grow  at 
different  rates  (Fig.  56) ;  and  changes  in  the  ratio  between  the  two 
go  a  long  way  to  explain  the  differences  in  shape  between  one  dog's 
skull  and  another's,  between  the  greyhound's  and  the  pug's.  But 
using  Huxley's  own  data  (after  Becher)  for  the  sheepdog,  I  find  the 
ratio  between  the  facial  and  cranial  portions  of  the  skull  to  be,  once 
again,  a  simple  linear  one. 

Measurements  of  skull  of  sheep-dog  (30  specimens)  *  {mm.) 

Mean  length  of  facial  region  (y) 

Mean  length  of  cranial  region- (x) 

Calculated  values  for  cranial 
region:  x  =  22-7 +0-88y 

And  now,  returning  to  the  fiddler-crab,  we  find  that  after  the 
crab  has  reached  a  certain  size  and  the  first  phase  of  rapid  growth 
is  over,  claw  and  body  grow  in  simple  linear  relation  to  one  another, 
and  the  heterogonic  or  compound-interest  formula  is  no  longer 
required: 

Fiddler-crab  (Uca  pugnax):  ratio  of  growth-rates ^  in  later  stages ^ 
of  claw  and  body  (mgm.) 

Weight  of  body  less      872      983       1080      1165      1212      1291       1363       1449 
claw  {x) 

Weight  of  large  418      461         537        594        617        670        699        778 

claw  {y) 

Do.,  calculated:  413       480        538        590        617        665        708         759 

y=0-6a;-110 

*  Data  from  A.  Becher,  in  Archivf.  Naturgesch.  (A),  1923;  see  Huxley,  Problems 
of  Relative  Growth,  p.  18,  and  Biol.  Centralbl.  loc.  cit.  Here,  and  in  the  previous 
case  of  Cydommatus,  the  equation  has  been  arrived  at  in  a  very  simple  way.  Take 
any  two  values,  x^,  x^,  and  the  corresponding  values,  yi,  y^.    Then  let 

x-Xj^  ^  y-yi 
^-^  Vz-Vx 
Of -65-3  v-48-3 


e.g. 

or 

from  which  a;  =  22-7+0-88y. 

We  may  with  advantage  repeat  this  process  with  other  values  of  x  and  y;  and 
take  the  mean  of  the  results  so  obtained. 


112-6-65-3      1020-48-3^ 

a; -653  _y-48-3 
47-3     "■     53-7    * 


TIT] 


OF  HETEROGONY 


211 


Once  again  we  find  close  agreement  between  the  observed  and 
calculated  values,  although  the  observations  are  somewhat  few  and 
the  equation  is  arrived  at  in  a  simple  way.  We  may  take  it  as 
proven  that  the  relation  between  the  two  growth-rates  is  essentially 
linear. 

A  compound-interest  law  of  growth  occurs,  as  Malthus  knew, 
in  cases,  and  at  times,  of  rapid  and  unrestricted  growth.  But 
unrestricted  growth  occurs  under  special  conditions  and  for  brief 


50  100 

Length  of  cranial  portion  (mm.) 


150 


Fig.  56.     Relative  growth  of  the  cranial  and  facial  portions  of  the 
skull  in  the  sheepdog.    Cf.  Huxley,  p.  18,  after  Becher, 


periods;  it  is  the  exception  rather  than  the  rule,  whether  in  a 
population  or  in  the  single  organism.  In  cases  of  diiferential 
growth  the  compound-interest  law  manifests  itself,  for  the  same 
reason,  when  one  of  the  two  growth-rates  is  rapid  and  "unre- 
stricted," and  when  the  discrepancy  between  the  two  growth-rates  is 
consequently  large,  for  instance  in  the  fiddler-crabs.  The  compound- 
interest  law  is  a  very  natural  mode  of  growth,  but  its  range  is 


212 


THE  RATE  OF  GROWTH 


[CH. 


limited.     A  linear  relation,  or  simple-interest  law,  seems  less  likely 
to  occur;   but  the  fact  is,  it  does  occur,  and  occurs  commonly. 

On  so-called  dimorphism 

In  a  well-known  paper,  Bateson  and  Brindley  shewed  that  among 
a  large  number  of  earwigs  collected  in  a  particular  locality,  the 
males  fell  into  two   groups,   characterised   by  large  or  by  small 


DD 


Fig.  57.     Tail-forceps  of  earwig.     From  Martin  Burr,  after  Willi  Kuhl. 
150 


^  100 


mm.  10 


Length  in  mm. 

Fig.  .58.     Variability  of  length  of  tail-forceps  in  a  sample  of  earwigs. 

After  Bateson  and  Brindley,  P.Z.S.  1892,  p.  588. 

tail-forceps  (Fig.  57),  with  few  instances  of  intermediate  magnitude*. 
This  distribution  into  two  groups,  according  to  magnitude,  is 
illustrated    in    the    accompanying    diagram    (Fig.    58);    and    the 

*  W.  Bateson  and  H.  H.  Brindley,  On  some  cases  of  variation  in  secondary 
sexual  characters  [Forficula,  Xylotrupa],  statistically  examined,  P.Z.S.  1892, 
pp.  585-594.  Cf.  D.  M.  Diakonow,  On  dimorphic  variability  of  Forficula,  Joum. 
Genet,  xv,  pp.  201-232,  1925;  and  Julian  Huxley,  The  bimodal  cephalic  horn  of 
Xylotrupa,  ibid,  xviii,  pp.  45-53,  1927. 


Ill 


OF  DIMORPHIC  GROWTH 


213 


phenomenon  was  described,  and  has  been  often  quoted,  as  one 
of  dimorphism  or  discontinuous  variation.  In  this  diagram  the 
time-element  does  not  appear;  but 'it  looks  as,  though  it  lay  close 
behind.  For  the  two  size-groups  into  which  the  tails  of  the  earwigs 
fall  look  curiously  hke  two  age-groups  such  as  we  have  already 
studied  in  a  fish,  where  the  ages  and  therefore  also  the  magnitudes  of  a 
random  sample  form  a  discontinuous  series  (Fig.  59).  And  if,  instead 
of  measuring  the  whole  length  of  our  fish,  we  had  confined  ourselves 
to  particular  parts,  .such  as'  head,  or  tail  or  fin,  we  should  have 
obtained  discontinuous  curves  of  distribution  for  the  magnitudes 


200 


150 


100 


Ocm.  \5  20  25  30 

Length  in  cm. 
Fig.  59.     Length  of  body  in  a  random  sample  of  plaice. 

of  these  organs,  just  as  for  the  whole  body  of  the  fish,  and  just  as 
for  the  tails  of  Bateson's  earwigs.  The  differences,  in  short,  with 
which  Bateson  was  dealing  were  a  question  of  magnitude,  and  it 
was  only  natural  to  refer  these  diverse  magnitudes  to  diversities  of 
growth;  that  is  to  say,  it  seemed  natural  to  suppose  that  in  this 
case  of  "dimorphism,"  the  tails  of  the  one  group  of  earwigs  (which 
Bateson  called  the  "high  males")  had  either  grown  faster,  or  had 
been  growing  for  a  longer  period  of  time,  than  those  of  the  "low 
males."  If  the  whole  random  sample  of  earwigs  were  of  one  and 
the  same  age,  the  dimorphism  would  appear  to  be  due  to  two 
alternative  values  for  the  mean  growth-rate,  individual  earwigs 
varying  around  one  mean  or  the  other.     If,  on  the  other  hand,  the 


214 


THE  RATE  OF  GROWTH 


[CH. 


two  groups  of  earwigs  were  of  diiferent  ages,  or  had  passed  through 
one  moult  more  or  less,  the  phenomenon  would  be  simple  indeed, 
and  there  would  be  no  more  to  be  said  about  it*.  Diakonow  made 
the  not  unimportant  observation  that  in  earwigs  living  in  un- 
favourable conditions  only  the  short-tailed  type  tended  to  appear. 
In  apparent  close  analogy  with  the  case  of  the  earwigs,  and  in 
apparent  corroboration  of  their  dimorphism  being  due  to  age, 
Fritz  Werner  measured  large  numbers  of  water-fleas,  all  apparently 
adult,  found  his  measurements  falling  into  groups  and  so  giving 
multimodal  curves.  The  several  cusps,  or  modes,  he  interpreted 
without  difficulty  as  indicating  diiferences  of  age,  or  the  number 
of  moults  which  the  creatures  had  passed  through f  (Fig.  BO). 


9 
/i.  60 


10    II    12   13   14  15    16    17   18   19  20  21    22  23  24 

80  100  120  140  160/1 

Length  in  ^ 

Fig.  60.     Measurements  of  the  dorsal  edge  in  a  population  of 
Chydorus  sphaericus,  a  water-flea. 


From  Fritz  Werner. 


An  apparently  analogous  but  more  difficult  case  is  that  of  a 
certain  little  beetle,  Onthofhagus  taurus,  which  bears  two  "horns" 
on  its  head,  of  variable  size  or  prominence.  Linnaeus  saw  in  it 
a  single  species,  Fabricius  saw  two;  and  the  question  long  remained 
an  open  one  among  the  eniomologists.  We  now  know  that  there 
are  two  "modes,"  two  predominant  sizes  in  a  continuous  range  of 

*  The  number  of  moults  is  known  to  be  variable  in  many  species  of  Orthoptera, 
and  even  occasionally  in  higher  insects;  and  how  the  number  of  moults  may  be 
influenced  by  hunger,  damp  or  cold  is  discussed  by  P.  P.  Calvert,  Proc.  Amer. 
Pkilos.  Soc.  Lxviii,  p.  246,  1929.  On  the  number  of  moults  in  earwigs,  see  E.  B. 
Worthington,  Entomologist,  1926,  and  W.  K.  Weyrauch,  Biol.  Centralbl.  1929, 
pp.  543-5^8. 

■f  Fritz  Werner,  Variationsanalytische  Untersuchungen  an  Chydoren,  Ztschr.  f. 
Morphologie  u.  Oekologie  d.  Tiere,  n,  pp.  58-188,  1924. 


Ill] 


OF  DIMORPHIC  GROWTH 


215 


variation*.  In  the  "complete  metamorphosis"  of  a  beetle  there 
is  no  room  for  a  moult  more  or  less,  and  the  reason  for  the  two 
modal  sizes  remains  hidden  (Fig.  61). 

But  new  hght  has  been  thrown  on  the  case  of  the  earwigs,  which 
may  help  to  explain  other  obscure  diversities  of  shape  and  size 
within  the  class  of  insects.  At  metamorphosis,  and  even  in  a  simple 
moult,  the  external  organs  of  an  insect  may  often  be  seen  to  unfold, 
as  do,  for  instance,  the  wings  of  a  butterfly;  they  then  quickly 
harden,  in  a  form  and  of  a  size  with  which  ordinary  gradual  growth 

I 


200 


Head  of  0.'tauru8;two 
forms  of  maJe 


^N 


A. 


0  0*5  1-0 

Length  of  horn  (mm.) 
Fig.  61.    Two  forms  of  the  male,  in  the  beetle  Ontkophagvs  taurus. 

has  had  nothing  directly  to  do.  This  is  a  very  peculiar  phenomenon, 
and  marks  a  singular  departure  from  the  usual  interdependence  of 
growth  and  form.  When  the  nymph,  or  larval  earwig,  is  about  to 
shed  its  skin  for  the  last  time,  the  tail-forceps,  still  soft  and  tender, 
are  folded  together  and  wrapped  in  a  sheath;  they  need  to  be 
distended,  or  inflated,  by  a  combined  pressure  of  the  body-fluid 
(or  haemolymph)  and  an  intake  of  respiratory  air.     If  all  goes  well, 

*  Rene  Paulian,  Bull.  Soc.  Zool.  Fr.  1933;    also  Le  polymorpMsme  des  mMea  de 
Cddoptires,  Paris,  1935,  p.  8. 


216  THE  RATE  OF  GROWTH  [ch. 

the  forceps  expand  to  their  full  size;  if  the  .reature  be  weak  or 
underfed,  inflation  is  incomplete  and  tho  tail-forceps  remain  small. 
In  either  case  it  is  an  affair  of  a  few  critical  moments  during  the 
final  ecdysis;  in  ten  minutes  or  less,  the  chitin  has  hardened,  and 
shape  and  size  change  no-  more.  Willi  Kuhl,  who  has  given  us  this 
interesting  explanation,  suggests  that  the  dimorphism  observed  by 
Bateson  and  by  Diakonow  is  not  an  essential  part  of  the  pheno- 
menon; he  has  found  it  in  one  instance,  but  in  other  and  much  larger 
samples  he  has  found  all  gradations,  but  only  a  single,-  well-marked 
unimodal  peak*. 

The  effect  of  temperature'f 

The  rates  of  growth  which  we  have  hitherto  dealt  with  are  mostly 
based  on  special  investigations,  conducted  under  particular  local 
conditions;  for  instance,  Quetelet's  data,  so  far  as  we  have  used 
them  to  illustrate  the  rate  of  growth  in  man,  are  drawn  from  his 
study  of  the  Belgian  people.  But  apart  from  that  "fortuitous" 
individual  variation  which  we  have  already  considered,  it  is  obvious 
that  the  normal  rate  of  growth  will  be  found  to  vary,  in  man  and 
in  other  animals,  just  as  the  average  stature  varies,  in  different 
locahties  and  in  different  "races."  This  phenomenon  is  a  very 
complex  one,  and  is  doubtless  a  resultant  of  many  undefined  con- 
tributory causes;  but  we  at  least  gain  something  in  regard  to  it 
when  we  discover  that  rate  of  growth  is  directly  affected  by 
temperature,  and  doubtless  by  other  physical  conditions.  Reaumur 
was  the  first  to  shew,  and  the  observation  was  repeated  by  Bonnet  {, 
that  the  rate  of  growth  or  development  of  the  chick  was  dependent 
on  temperature,  being  retarded  at  temperatures  below  and  somewhat 

*  Willi  Kuhl,  Die  Variabilitat  der  abdominalen  Korperanhange  bei  Forficula, 
Ztsch.  Morph.  u.  Oek.  d.  Tiere,  xii,  p.  299,  1924.  Cf.  Malcolm  Burr,  Discovery, 
1939,  pp.  340^345. 

t  The  temperature  limitations  of  life,  and  to  some  extent  of  growth,  are  sum- 
marised for  a  large  number  of  species  by  Davenport,  Exper.  Morphology,  cc.  viii, 
xviii,  and  by  Hans  Przibram,  Exp.  Zoologie,  iv,  c.  v.^ 

X  Reaumur,  L'art  de  faire  eclorre  et  elever  en  toute  saison  des  oiseaux  domestiques, 
soil  par  le  moyen  de  la  chaleur  du  fumee,  soil  par  le  moyen  de  celle  du  feu  ordinaire, 
Paris,  1749.  He  had  also  studied,  a  few  years  before,  the  effects  of  heat  and  cold 
on  growth-rate  and  duration  of  life  in  caterpillars  and  chrysalids:  Memoires,  ii, 
p.  1,  de  la  dnree  de  la  vie  des  crisalides  (1736).  See  also  his  Observations  du 
Thermometre,  etc.,  Mem.  Acad.,  Paris,  1735,  pp.  345-376. 


Ill]  THE  EFFECT'  OF  TEMPERATURE  217 

accelerated  at  temperatures  above  the  normal  temperature  of 
incubation,  that  is  to  say  the  temperature  of  the  sitting- hen.  In 
the  case  of  plants  the  fact  that  growth  is  greatly  affected  by  tem- 
perature is  a  matter  of  famiUar  knowledge;  the  subject  was  first 
carefully  studied  by  Alphonse  De  Candolle,  and  his  results  and  those 
of  his  followers  are  discussed  in  the  textbooks  of  botany*. 

That  temperature  is  only  one  of  the  climatic  factors  determining  growth  and 
yield  is  well  known  to  agriculturists;  and  a  method  of  "multiple  correlation" 
has  been  used  to  analyse  the  several  influences  of  temperature  and  of  rainfall 
at  different  seasons  on  the  future  yield  of  our  own  crops  f.  The  same  joint 
influence  can  be  recognised  in  the  bamboo;  for  it  is  said  (by  Lock)  that  the 
growth-rate  of  the  bamboo  in  Ceylon  is  proportional  to  the  humidity  of  the 
atmosphere,  and  again  (by  Shibata)  that  it  is  proportional  to  the  temperature 
in  Japan.  But  BlackmanJ  suggests  that  in  Ceylon  temperature  conditions 
are  all  that  can  be  desired,  but  moisture  is  apt  to  be  deficient,  while  in  Japan 
there  is  rain  in  abundance  but  the  average  temperature  is  somewhat  low: 
so  that  in  the  one  country  it  is  the  one  factor,  and  in  the  other  country  the 
other,  whose  variation  is  both  conspicuous  and  significant.  After  all,  it  is 
probably  rate  of  evaporation,  the  joint  result  of  temperature  and  humidity, 
which  is  the  crux  of  the  matter§.  "Climate"  is  a  subtle  thing,  and  includes 
a  sort  of  micro-meteorology.  A  sheltered  corner  has  a  climate  of  its  own;  one 
side  of  the  garden -wall  has  a  different  climate  to  the  other;  and  deep  in  the 
undergrowth  of  a  wood  celandine  and  anemone  enjoy  a  climate  many  degrees 
warmer  than  what  is  registered  on  the  screen ||. 

Among  the  mould-fungi  each  several  species  has  its  own  optimum  tempera- 
ture for  germination  and  growth.  At  this  optimum  temperature  growth  is 
further  accelerated  by  increase  of  humidity ;  and  the  further  we  depart  from 
the  optimum  temperature,  the  narrower  becomes  the  range  of  humidity  within 
which  growth  can  proceed^.  Entomologists  know,  in  like  manner,  how  over- 
abundance of  an  insect-pest  comes,  or  is  apt  to  come,  with  a  double  optimum 
of  temperature  and  humidity. 

*  Cf.  {int.  at.)  H.  de  Vries,  Materiaux  pour  la  connaissance  de  I'influence  de  la 
temperature  sur  les  plantes,  Arch.  Neerlandaises,  v,  pp.  385-401,  1870;  C.  Linsser, 
Periodische  Erscheinungen  des  Pflanzenlebens,  Mem.  Acad,  des  Sc,  8t  Petersbourg 
(7),  XI,  XII,  1867-69;  Koppen,  Warme  und  Pflanzenwachstum,  Bull.  Soc.  Imp. 
Nat.,  Moscou,  xliii,  pp.  41-110,  1871;  H.  Hoffmann,  Thermisehe  Vegetations- 
constanten,  Ztschr.  Oesterr.  Ges.  f.  Meteorologie,  xvii,  pp.  121-131,  1881;  Pheno- 
logische  Studien,  Meteorolog.  Ztschr.  iii,  pp.  H3-120,  1886. 

t  See  [int.  al.)  R.  H.  Hooker,  Journ.  Roy.  Statist.  Soc.  1907,  p.  70;  Journ.  Roy. 
Meteor.  Soc.  1922,  p.  46. 

I  F.  F.  Blackman,  Ann.  Bot.  xix,  p.  281,  1905. 

§   Szava-Kovatz,  in  Petermann's  Mitteilungen,  1927,  p.  7. 

II  Cf.  E.  J.  Salisbury,  On  the  oecological  aspects  of  Meteorology,  Q.J.R.  Meteorol. 
Soc.  July  1939. 

^  R.  G.  Tomkins,  Proc.  R.S.  (B),  cv,  pp.  375^01,  1929. 


218 


THE  RATE  OF  GROWTH 


[CH. 


The  annexed  diagram  (Fig.  62),  showing  growth  in  length  of  the 
roots  of  some  common  plants  at  various  temperatures,  is  a  sufl&cient 
illustration  of  the  phenomenon.  We  see  that  there  is  always  a 
certain  temperature  at  which  the  rate  is  a  maximum ;  while  on  either 
side  of  the  optimum  the  rate  falls  off,  after  the  fashion  of  the  normal 
curve  of  error.  We  see  further,  from  the  data  given  by  Sachs  and 
others,  that  the  optimum  is  very  much  the  same  for  all  the  common 
plants  of  our  own  chmate.    For  these  it  is  somewhere  about  26°  C. 


80 


70 


M^ie  16  20  22  24  26  28  30  32  34  36  38  40° 

Temp. 

Fig.  62.    Relation  of  rate  of  growth  to  temperature  in  certain 

plants.     From  Sachs's  data. 


(say  77°  F.),  or  about  the  temperature  of  a  warm  summer's  day; 
while  it  is  considerably  higher,  naturally,  in  such  plants  as  the  melon 
ot*  the  maize,  which  are  at  home  in  warmer  countries  than  our  own. 
The  bacteria  have,  in  like  manner,  their  various  optima,  and  some- 
times a  high  one.  The  tuberculosis-bacillus,  as  Koch  shewed,  only 
begins  to  grow  at  about  28°  C,  and  multipUes  most  rapidly  at 
37-38°,  the  body- temperature  of  its  host. 

The  setting  and  ripening  of  fruit  is  a  phase  of  growth  stiU  more 
dependent  on  temperature;  hence  it  is  a  "dehcate  test  of  climate," 
and  a  proof  of  its  constancy,  that  the  date-palm  grows  but  bears 


Ill]  THE  EFFECT  OF  TEMPERATURE  219 

no  fruit  in  Judaea,  and  the  vine  bears  freely  at  Eshcol,  but  not  in 
the  hotter  country  to  the  south*.  Shellfish  have  their  own  appro- 
priate spawning-temperatures;  it  needs  a  warm  summer  for  the 
oyster  to  shed  her  spat,  and  Hippopus  and  Tridacna,  the  great  clams 
of  the  coral-reefs,  only  do  so  when  the  water  has  reached  the  high 
temperature  of  30°  C.  For  brown  trout,  6°  C.  is  found  to  be  a 
critical  temperature,  a  minimum  short  of  which  they  do  not  grow 
at  all;  it  follows  that  in  a  Highland  burn  their  growth  is  at  a 
standstill  for  fully  half  the  yearf. 

That  a  rise  of  temperature  accelerates  growth  is  but  part  of  the 
story,  and  is  not  always  true.  Several  insects,  experimentally 
reared,  have  been  found  to  diminish  in  size  as  the  temperature 
increased  J;  and  certain  flies  have  been  found  to  be  larger  in  their 
winter  than  their  summer  broods.  The  common  copepod,  Calmius 
finmarchicus,  has  spring,  summer  and  autumn  broods,  which  (at 
Plymouth §)  are  large,  middle-sized  and  small;  but  the  large  spring 
brood  are  hatched  and  reared  in  the  cold  "winter"  water,  and  the 
small  autumn-winter  brood  in  the  warmest  water  of  the  year.  In 
the  cold  waters  of  Barents  Sea  Calanus  grows  larger  still;  of  an 
allied  genus,  a  large  species  lives  in  the  Antarctic,  a  small  one  in 
the  tropics,  a  middle-sized  is  common  in  the  temperate  oceans. 
The  large  size  of  many  Arctic  animals,  coelenterates  and  crustaceans, 
is  well  known;  and  so  is  that  of  many  tropical  forms,  Hke  Fungia 
among  the  corals,  or  the  great  Tritons  and  Tridacnas  among 
molluscs.  Another  common  phenomenon  is  the  increasing  number 
of  males  in  late  summer  and  autumn,  as  in  the  Rotifers  and  in  the 
above-mentioned  Calani.  All  these  things  seem  somehow  related 
to  temperature;  but  other  physical  conditions  enter  iilto  the  case, 
for  instance  the  amount  of  dissolved  oxygen  in  the  cold  waters,  and 
the  physical  chemistry  of  carbonate  of  Hme  in  the  warm||. 

The  vast  profusion  of  life,  both  great  and  small,  in  Arctic  seas,  the  multitude 
of  individuals  and  the  unusual  size  to  which  many  species  grow,  has  been 
often  ascribed  to  a  superabundance  of  dissolved  oxygen,  but  oxygen  alone 
would  not  go   far.     The   nutrient  salts,   nitrates   and  phosphates,   are   the 

*  Cf.  J.  W.  Gregory,  in  Geogr.  Journ.  1914,  and  Journ.  E.  Geogr.  Soc.  Oct.  1930. 

t  Cf.  C.  A.  Wingfield,  op.  cit.  supra,  p.  176. 

X  B.  P.  Uvarow,  Trans.  Ent.  Soc.  Lond.  lxxix,  p.  38,  1931. 

§   W.  H.  Golightly  and  LI.  Lloyd,  in  Nature,  July  22,  1939. 

li   Cf.  B.  G.  Bogorow  and  others,  in  the  Journ.  M.B.A.  xix,  1933-34. 


220  THE  RATE  OF  GROWTH  [ch. 

limiting  factor  in  the  growth  of  that  micro-vegetation  with  which  the  whole 
cycle  of  life  begins.  The  tropical  oceans  are  often  very  bare  of  these  salts; 
in  our  own  latitudes  there  is  none  too  much,  and  the  spring-growth  tends  to 
use  up  the  supply.  But  we  have  learned  from  the  Discovery  Expedition 
that  these  salts  are  so  abundant  in  the  Antarctic  that  plant-growth  is  never 
checked  for  stint  of  them.  Along  the  Chilean  coast  and  in  S.W.  Africa, 
cold  Antarctic  water  wells  up  from  below  the  warm  equatorial  current.  It 
is  ill-suited  for  the  growth  of  corals,  which  build  their  reefs  in  the  warmer 
waters  of  the  eastern  side ;  but  it  teems  with  nourishment,  breeds  a  plankton- 
fauna  of  the  richest  kind,  which  feeds  fishes  preyed  on  by  innumerable  birds, 
the  guano  of  which  is  sent  all  over  the  world.  Now  and  then  persistent  winds 
thrust  the  cold  current  aside ;  a  new  warm  current,  el  Nino  of  the  Chileans, 
upsets  the  old  equilibrium ;  the  fishes  die,  the  water  stinks,  the  birds  starve. 
The  same  thing  happens  also  at  Walfisch  Bay,  where  on  such  rare  occasions 
dead  fish  lie  piled  up  high  along  the  shore. 

It  is  curiously  characteristic  of  certain  physiological  reactions, 
growth  among  them,  to  be  affected  not  merely  by  the  temperature 
of  the  moment,  but  also  by  that  to  which  the  organism  has  been 
previously  and  temporarily  exposed.  In  other  words,  acclimatisation 
to  a  certain  temperature  may  continue  for  some  time  afterwards  to 
affect  all  the  temperature  relations  of  the  body*.  That  temporary 
c<!>ld  may,  under  certain  circumstances,  cause  a  subsequent  accelera- 
tion of  growth  is  made  use  of  in  the  remarkable  process  known  as 
vernalisation.  An  ingenious  man,  observing  that  a  winter  wheat  failed 
to  flower  when  sown  in  spring,  argued  that  exposure  to  the  cold  of 
winter  was  necessary  for  its  subsequent  rapid  growth ;  and  this  he 
verified  by  "  chiUing"  his  seedlings  for  a  month  to  near  freezing-point, 
after  which  they  grew  quickly,  and  flowered  at  the  same  time  as  the 
spring  wheat.  The  economic  advantages  are  great  of  so  shortening 
the  growing  period  of  a  crop  as  to  protect  it  from  autumn  frosts  in  a 
cold  chmate  or  summer  drought  in  a  hot  one ;  much  has  been  done, 
especially  by  Lysenko  in  Russia,  with  this  end  in  viewf. 

The  most  diverse  physiological  processes  may  be  afl'ected  by 
temperature.  A  great  astronomer  at  Mount, Wilson,  in  California, 
used  some  idle  hours  to  watch  the  "trail-running"  ants,  which  run 
all  night  and  all  day.  Their  speed  increases  so  regularly  with  the 
temperature  that  the  time  taken  to  run  30  cm.  suffices  to  tell  the 

*  Cf.    Kenneth   Mellanby,    On    temperature   coefficients    and    acclimatisation, 
Nature,  3  August  1940. 
t  Of.  {int.  al.)  V.  H.  Blackman,  in  Nature,  June  13,  1936. 


Ill]  THE  TEMPERATURE  COEFFICIENT  221 

temperature  to  1°  C. !  Of  two  allied  species,  one  ran  nearly  half  as 
fast  again  as  the  other,  at  the  same  temperature*. 

While  at  low  temperatures  growth  is  arrested  and  at  temperatures 
unduly  high  hfe  itself  becomes  impossible,  we  have  now  seen  that 
within  the  range  of  more  or  less  congenial  temperatures  growth 
proceeds  the  faster  the  higher  the  temperature.  The  same  is  true 
of  the  ordinary  reactions  of  chemistry,  and  here  Van't  Hoff  and 
Arrheniusf  have  shewn  that  a  definite  increase  in  the  velocity  of 
the  reaction  follows  a  definite  increase  of  temperature,  according  to 
an  exponential  law:  such  that,  for  an  interval  of  n  degrees  the 
velocity  varies  as  x",  x  being  called  the  "temperature  coefficient" 
for  the  reaction  in  question  J.  The  law  holds  good  throughout  a 
considerable  range,  but  is  departed  from  when  we  pass  beyond 
certain  normal  limits ;  moreover,  the  value  of  the  coefficient  is  found 
to  keep  to  a  certain  order  of  magnitude — somewhere  about  2  for 
a  temperature-interval  of  10°  C. — which  means  to  say  that,  the 
velocity  of  the  reaction  is  just  about  doubled,  more  or  less,  for  a 
rise  of  10°  C. 

This  law,  which  has  become  a  fundamental  principle  of  chemical 
mechanics,  is  applicable  (with  certain  qualifications)  to  the  pheno- 
mena of  vital  chemistry,  as  Van't  Hoff  himself  was  the  first  to  declare ; 
and  it  follows  that,  on  much  the  same  fines,  one  may  speak  of  a 
"temperature  coefficient"  of  growth.  At  the  same  time  we  must 
remember  that  there  is  a  very  important  difference  (though  we  need 
not  call  it  a  fundamental  one)  between  the  purely  physical  and  the 

*  Harlow  Shapley,  On  the  thermokinetics  of  Dolichoderine  ants,  Proc.  Nat. 
Acad.  Sci.  x,  pp.  436-439,  1924. 

t  Van't  HofF  and  Cohen,  Studien  zur  chemischen  Dynamik,  1896;  Sv.  Arrhenius, 
Ztschr.  f.  phys.  Chemie,  iv,  p.  226. 

X  For  various  instances  of  a  temperature  coefficient  in  physiological  processes, 
see  (e.g.)  Cohen,  Physical  Chemistry  f or ...  Biologists  (Enghsh  edition),  1903; 
Kanitz  and  Herzog  in  Zeitschr.  f.  Elektrochemie,  xi,  1905;  F.  F.  Blackman,  Ann. 
Bot.  XIX,  p.  281,  1905;  K.  Peter,  Arch.f.  Entw.  Mech.  xx,  p.  130,  1905;  Arrhenius, 
Ergebn.  d.  Physiol,  vii,  p.  480,  1908,  and  Quantitative  Laws  in  Biological  Chemistry, 
1915;  Krogh  in  Zeitschr.  f.allgem.  Physiologie,xyi, -pp.  163,178,1914;  James  Gray, 
Proc.  E.S.  (B),  xcv,  pp.  6-15,  1923;  W.  J.  Crozier,  many  papers  in  Journ.  Gen. 
Physiol.  1924;  J.  Belehradek,  in  Biol.  Reviews,  v,  pp.  1-29,  1930.  On  the  general 
subject,  see  E.  Janisch,  Temperaturabhangigkeit  biologischer  Vorgange  und  ihrer 
kurvenmassige  Analyse,  Pfluger's  Archiv,  ccix,  p.  414,  1925;  G.  and  P.  Hertwig, 
Regulation  von  Wachstum . .  .  durch  Umweltsfaktoren,  in  Hdb.  d.  normal,  u.  pathol. 
Physiologie,  xvi,  1930. 


222  THE  RATE  OF  GROWTH  [ch. 

physiological  phenomenon,  in  that  in  the  former  we  study  (or  seek 
and  profess  to  study)  one  thing  at  a  time,  while  in  the  living  body 
we  have  constantly  to  do  with  factors  which  interact  and  interfere; 
increase  in  the  one  case  (or  change  of  any  kind)  tends  to  be  con- 
tinuous, in  the  other  case  it  tends  to  be  brought,  or  to  bring 
itself,  to  arrest.  This  is  the  simple  meaning  of  that  Law  of 
Optimum,  laid  down  by  Errera  and  by  Sachs  as  a  general  principle 
of  physiology ;  namely  that  every  physiological  process  which  varies 
(Hke  growth  itself)  with  the  amount  or  intensity  of  some  external 
influence,  does  so  under  such  conditions  that  progressive  increase  is 
followed  by  progressive  decrease;  in  other  words,  the  function  has 
its  optimum  condition,  and  its  curve  shews  a  definite  maximu^n. 
In  the  case  of  temperature,  as  Jost  puts  it,  it  has  on  the  one  hand 
its  accelerating  effect,  which  tends  to  follow  Van't  Hoff's  law.  But 
it  has  also  another  and  a  cumulative  effect  upon  the  organism: 
"Sie  schadigt  oder  sie  ermiidet  ihn,  und  je  hoher  sie  steigt  desto 
rascher  macht  sie  die  Schadigung  geltend  und  desto  schneller  scbxeitet 
sie  voran*."  It  is  this  double  effect  of  temperature  on  the  organism 
which  gives,  or  helps  to  give  us  our  "optim'im"  curves,  which  (like 
all  other  curves  of  frequency  or  error)  are  the  expression,  not  of  a 
single  solitary  phenomenon,' but  of  a  more  or  less  complex  resultant. 
Moreover,  as  Blackman  and  others  have  pointed  out,  our  "optimum" 
temperature  is  ill-defined  until  we  take  account  also  of  the  duration 
of  our  experiment;  for  a  high  temperature  may  lead  to  a  short  but 
exhausting  spell  of  rapid  growth,  while  the  slower  rate  manifested 
at  a  lower  temperature  may  be  the  best  in  the  end.  The  mile  and 
the  hundred  yards  are  won  by  different  runners;  and  maximum 
rate  of  worldng,^  and  maximum  amount  of  work  done,  are  two  very 
different  things  f. 

In  the  case  of  maize,  a  certain  series  of  experiments  shewed  that 
the  growth  in  length  of  the  roots  varied  with  the  temperature  as 
follows  J: 

*  On  such  limiting  factors,  or  counter-reactions,  see  Putter,  Ztschr.  f.  aUgem. 
Physiologic,  xvi,  pp.  574-627,  1914. 

t  Cf.  L.  Errera,  UOptimum,  1896  (Recueil  d'oeuvres,  Physiologie  genirale,  pp.  338- 
368,  1910) ;  Sachs,  Physiologie  d.  Pflanzen,  1882,  p.  233;  PfeflFer,  Pflanzenphysiologie, 
n,  p.  78,  194;  and  cf,  Jost,  Ueber  die  Reactionsgeschwindigkeit  ira  Organismus, 
Biol.  CentraWl.  xxvi,  pp.  225-244,  1906. 

t  After  Koppen,  Bull.  Soc.  Nat.  Moscou,  XLin,  pp.  41-101,  1871. 


Ill]  THE  TEMPERATURE  COEFFICIENT  223 


Temperature 

Growth  in  48  hours 

°C. 

mm. 

18-0 

11 

23-5 

10-8 

26-6 

29-6 

23-5 

26-5 

30-2 

64-6 

33-5 

69-5 

36-5 

20-7 

us 

write  our 

formula  in 

the  form 

V(>^) 

=  x^,    or  In 

laF..._^- 

\o0V  =  nA 

Then  choosing  two  values  out  of  the  above  experimental  series 
(say  the  second  and  the  second-last),  we  have  t  =  23-5,  n  =  10, 
and  F,  V  =  10-8  and  69-5  respectively. 

.         ,.     -             log  69-5  -  log  10-8      , 
Accordmgly,  — ^ — — =  log  x, 

0-8414  -  0-034 
or ^^  =  0-0808, 

and  therefore  the  temperature-coefficient 

=  antilog  0-0808  =  1-204  (for  an  interval  of  1°  C). 

This  first  approximation  might  be  much  improved  by  taking  account 
of  all  the  experimental  values,  two  only  of  which  we  have  yet  made 
use  of;  but  even  as  it  is,  we  see  by  Fig.  63  that  it  is  in  very  fair 
accordance  with  the  actual  results  of  observation,  within  those 
particular  limits  of  temperature  to  which  the  experiment  is  confined. 
For  an  experiment  on  Lupinus  albus,  quoted  by  Asa  Gray* 
I  have  worked  out  the  corresponding  coefficient,  but  a  httle  more 
carefully.  Its  value  I  find  to  be  1-16,  or  very  nearly  identical  with 
that  we  have  just  found  for  the  maize;  and  the  correspondence 
between  the  calculated  curve  and  the  actual  observations  is  now 
a  close  one. 

Miss  I.  Leitch  has  made  careful  observations  of  the  rate  of  growth  of  rootlets 
of  the  Pea;  and  I  have  attempted  a  further  analysis  of  her  principal  resultsf . 

*  Asa  Gray,  Botany,  p.  387. 

t  I.  Leitch,  Some  experiments  on  the  influence  of  temperature  on  the  rate 
of  growth  in  Pisum  sativum,  Ann.  Bot.  xxx,  pp.  25-46,  1916,  especially  Table  III, 
p.  45.  Cf.  Priestley  and  Pearsall,  Growth  studies,  Ann.  Bot.  xxxvi,  pp.  224-249, 
1922. 


224 


THE  RATE  OF  GROWTH 


[CH. 


In  Fig.  64  are  shewn  the  mean  rates  of  growth  (based  on  about  a  hundred 
experiments)  at  some  thirty-four  different  temperatures  between  0-8°  and 
29-3°,  each  experiment  lasting  rather  less  than  twenty-four  hours.  Working 
out  the  mean  temperature  coefficient  for  a  great  many  combinations  of  these 
values,  I  obtain  a  value  of  1-092  per  C.°,  or  2-41  for  an  interval  of  10°,  and 
a  mean  value  for  the  whole  series  shewing  a  rate  of  growth  of  just  about 
1  mm.  per  hour  at  a  temperature  of  20°.  My  curve  in  Fig.  64  is  drawn  from 
these  determinations;  and  it  will  be  seen  that,  while  it  is  by  no  means  exact 
at  the  lower  temperatures,  and  will  fail  us  altogether  at  very  high  tem- 
peratures, yet  it  serves  as  a  satisfactory  guide  to  the  relations  between  rate 
and  temperature  within  the  ordinary  limits  of  healthy  growth.     Miss  Leitch 


18       20        22        24        26         28        30        32       34°C 
Fig.  63.     Relation  of  rate  of  growth  to  temperature  in  maize.     Observed 
values  (after  Koppen),  and  calculated  curve. 

holds  that  the  curve  is  not  a  Van't  Hoff  curve ;  and  this,  in  strict  accuracy, 
we  need  not  dispute.  But  the  phenomenon  seems  to  me  to  be  one  into  which 
the  Van't  Hoff  ratio  enters  largely,  though  doubtless  combine'd  with  other 
factors  which  we  cannot  determine  or  eliminate. 

While  the  above  results  conform  fairly  well  to  the  law  of  the 
temperature-coefficient,  it  is  evident  that  the  imbibition  of  water 
plays  so  large  a  part  in  the  process  of  elongation  of  the  root  or 
stem  that  the  phenomenon  is  as  much  or  more  a  physical  than  a 
chemical  one:  and  on  this  account,  as  Blackman  has  remarked,  the 
data  commonly  given  for  the  rate  of  growth  in  plants  are  apt  to 
be    irregular,    and    sometimes    misleading*.     We    have    abundant 

*  F.  F.  Blackman,  Presidential  Address  in  Botany,  Brit.  Assoc.  Dublin,  1908. 


Ill] 


OF  TEMPERATURE  COEFFICIENTS 


225 


illustrations,  however,  among  animals,  in  which  we  may  study  the 
temperature-coefficient  under  circumstances  where,  though  the 
phenomenon  is  always  complicated,  true  metabohc  growth  or 
chemical  combination  plays  a  larger  role.  Thus  Mile.  Maltaux  and 
Professor  Massart*  have  studied  the  rate  of  division  in  a  certain 
flagellate,  Chilomonas  paramoecium,  and  found  the  process  to  take 
20 


/• 

— 

/ 

.7 

■~ 

7 

/ 

/ 

/ 

_ 

/ 

V 

- 

. 

/ 

/ 

— 

/ 

'/. 

/ 

/ 

1 — 

• 

/ 

/ 

^ 

— 

^i 

^^ 

• 

^ 

* 

- 

• 

^ 

y 

-^ 

• 
* 

"" 

per 
hour 

20 


1-8 

1-6 

1-4 

1-2 

1-0 

•8 

•6 

•4 

•2 


0  4°  8°        1 2°       1 6°       20°       24°  ^      28°      32°  ^ 

Fig,  64.     Relation  of  rate  of  growth  to  temperature  in  rootlets  of 
pea.     From  Miss  I.  Leitch's  data. 

29  minutes  at  15°  C,  12  at  25°,  and  only  5  minutes  at  35°  C.  These 
velocities  are  in  the  ratio  of  1 :  24:  5-76,  which  ratio  corresponds 
precisely  to  a  temperature-coefficient  of  24  for  each  rise  of  10°,  or 
about  1-092  for  each  degree  centigrade,  precisely  the  'Same  as  we 
have  found  for  the  growth  of  the  pea. 

By  means  of  this  principle  we  may  sometimes  throw  hght  on 
apparently  compUcated  experiments.     For  instance.  Fig.  65  is  an 

*  Rec.  de  VInst.  Bot.  de  Brzizelles,  vi,  1906. 


226 


THE  RATE  OF  GROWTH 


[CH. 


illustration,  which  has  been  often  copied,  of  0.  Hertwig's  work  on 
the  effect  of  temperature  on  the  rate  of  development  of  the  tadpole*. 


'1          ~fZi 

Z                   ^4^ 

1                   1       1 

/                    /         1 

~/      ~X  X- 

T_                 J-       L 

«' 

/                       /         1 

1                       I       1 

1            ill 

01      «^       .... 

-          t    1    14^ 

"^ 

-     4  -T  ttU- 

.    I  2^    riii 

.      t^'^     rtlf 

'^ 

v^      IJ  LE 

-  z^^     Tti-l^t 

M 

R  1  J^iItX- 

»^ 

VJV    ptj 

^  ^   t     Qj   3 

-l 

f-  /-    ziy  i 

f~ 

/^^'^  z  it-,/ 

"                                                 JJ- 

-i^'^-^Z   Jl   J 

\                                           JV    . 

t-  z  ^^zt'^ 

\  .,o^j>       y^  L/. 

%^^yjLt 

£^^^   V 

^        / 

^^^-""^^^^^^     .-^ 

>^  ""^^ 

'^'^^  ^c^^  -^^  ^^^ 

^^^^^^^^^     --' 

-■^ 

^  —  "  ^ — ::h^  "^  "^ 

j                                         TemneXcLl 

uie     Ceatign.a.d.e 

«*•  «•  2^"  «/• 


/*•  //"  It,'  I-.'  IV  li"  If  II'  lof  9"   a"    ^   ef   s' 


Fig.  65.  Diagram  shewing  time  taken  (in  days),  at  various  temperatures  {°  C), 
to  reach  certain  stages  of  development  in  the  frog:  viz.  I,  gastrula;  II, 
medullary  plate;  III,  closure  of  medullary  folds;  IV,  tail- bud;  V,  tail  and 
gills;  VI,  tail-fin;  VII,  operculum  beginning;  VIII,  do.  closing;  IX,  first 
appearance  of  hind-legs.     From  Jenkinson,  after  0.  Hertwig,  1898. 

*  0.  Hertwig,  Einfluss  der  Temperatur  auf  die  Entwicklung  von  Rana  fusca 
und  R.  esculenta,  Arch.  f.  mikrosk.  Anat.  li,  p.  319,  1898.  Cf.  also  K.  Bialaszewicz, 
Beitrage  z.  Kenntniss  d.  Wachsthumsvorgange  bei  Amphibienembryonen,  Ball. 
Acad.  Sci.  de  Cracovie,  p.  783,  1908;  Abstr.  in  Arch.  f.  Entwicklungsmech.  xxviii, 
p.  160,  1909:  from  which  Ernst  Cohen  determined  the  value  of  Q^q  (Vortrdge  iib. 
physikal.  C hemic  f.  Arzte,  1901;  English  edit.  1903). 


Ill] 


OF  TEMPERATURE  COEFFICIENTS 


227 


From  inspection  of  this  diagram,  we  see  that  the  time  taken  to 
attain  certain  stages  of  development  (denoted  by  the  numbers 
III-VII)  was  as  follows,  at  20°  and  at  10°  C,  respectively. 


At  20°  C. 

At  10°  C. 

Stage  III 

2-0 

6-5  days 

„      IV 

2-7 

8-1     „ 

„      V 

30 

10-7     „ 

;,  VI 

4-0 

13-5     „ 

„      VII 

50 

16-8     ,. 

Total 


16-7 


55-6 


25'C.        20°  15°  10°  5' 

Fig.  66.     Calculated  values,  corresponding  to  preceding  figure. 

That  is  to  say,  the  time  taken  to  produce  a  given  result  at  10° 
was  (on  the  average)  somewhere  about  55-6/16-7,  or  3-33,  times  as 
long  as  was  required  at  20°  C. 

We  may  then  put  our  equation  in  the  simple  form, 

x^^  =  3-33. 
Or,  10  log  X  =  log  3-33  =  0-52244. 

Therefore  log  x  =  0-05224, 

and  X  =  1-128. 


228  THE  RATE  OF  GROWTH  [ch. 

That  is  to  say,  between  the  intervals  of  10°  and  20°  C,  if  it  take 
m  days,  at  a  certain  given  temperature,  for  a  certain  stage  of 
development  to  be  attained,  it  will  take  m  x  1-128"  days,  when  the 
temperature  is  n  degrees  less,  for  the  same  stage  to  be  arrived  at. 

Fig.  66  is  calculated  throughout  from  this  value;  and  it  will  be 
found  extremely  concordant  with  the  original  diagram,  as  regards 
all  the  stages  of  development  and  the  whole  range  of  temperatures 
shewn;  in  spite  of  the  fact  that  the  coefficient  on  which  it  is  based 
was  derived  by  an  easy  method  from  a  very  few  points  on  the 
original  curves.  In  hke  manner,  the  following  table  shews  the 
"incubation  period"  for  trout-eggs,  or  interval  between  fertihsation 
and  hatching,  at  different  temperatures  * : 

Incubation-period  of  trout-eggs 


Temperature 

Days'  interval 

°C. 

before  hatching 

2-8 

165 

3-6 

135 

3-9 

121 

4-5 

109 

50 

103 

5-7 

96 

6-3 

89 

6-6 

81 

7-3 

73 

8-0 

65 

90 

56 

10-0 

47 

111 

38 

12-2 

32 

Choosing  at  random  a  pair  of  observations,  viz.  at  3-6°  and  10°, 
and  proceeding  as  before,  we  have 

10°  -  3-6°  =  64°. 

Then  (64)  =  '-g, 

or  6-4  X  log  X  =  log  135  —  log  47 

=  2-1303  -  1-6721  =  0-4582 
and  log  X  =  0-4582  -  6-4       =  0-0716, 

X  =  1-179. 

*  Data  from  James  Gray,  The  growth  of  fish,  Journ.  Exper.  Biology,  vi,  p.  126, 
1928. 


Ill]  OF  TEMPERATURE  COEFFICIENTS  229 

Using  three  other  pairs  of  observations,  we  have  the  following 
concordant  results : 

At  12-2°  and  2-8^,  x  -  M91 

10-0°         3-6°  M79 

9-0°         5-7°;         M78 

8-0°         5-0°  M65 

Mean  M8 

A  very  curious  point  is  that  (as  Gray  tells  us)  the  young  fish  which 
have  hatched  slowly  at  a  low  temperature  are  bigger  than  those 
whose  growth  has  been  hastened  by  warmth. 

Again,  plaice-eggs  were  found  to  hatch  and  grow  to  a  certain 
length  (4-6  mm.),  as  follows*: 

Temperature  (°  C.)  Days 

41  230 

6-1  181 

8-0  13-3 

10-1  10-3 

120  8-3 

From  these  we  obtain,  as  before,  the  following  constants: 

At  12°     and  8°,    x  =  M3 

12°  4-1°         M4 

10-1°         6-1°         M5 

8-0°         4-1°         M5 

Mean  M4 

The  value  of  x  is  much  the  same  for  the  one  fish  as  for  the  other. 

Karl  Peter  t,  experimenting  on  echinoderm  eggs,  and  making  use 
also  of  Richard  Hert wig's  experiments  on  young  tadpoles,  gives  the 
temperature-coefficients  for  intervals  of  10°  C.  (commonly  written 
Qio)  as  follows,  to  which  I  have  added  the  corresponding  values 
forg,: 

Sphaerechinus    Qiq  =2-15    Qi^  1-08 
Echinus  ?'13  1-08 

Rana  2-86  Ml 

*  Data  from  A.  C.  Johansen  and  A.  Krogh,  Influence  of  temperature,  etc., 
Publ.  de  Circonstance,  No.  68,  1914.  The  function  is  here  said  to  be  a  linear  one — 
which  would  have  been  an  anomalous  and  unlikely  thing". 

t  Der  Grad  der  Beschleunigung  tierischer  Entwicklung  durch  erhohte  Tem- 
peratur,  Arch.  f.  Entw.  Mech.  xx,  p.  130,  1905.  More  recently  Bialaszewicz  has 
determined  the  coefficient  for  the  rate  of  segmentation  in  Rana  as  being  2-4  per  10°  C. 


230  THE  RATE  OF  GROWTH  fcH. 

These  values  are  not  only  concordant,  but  are  of  the  same  order 
of  magnitude  as  the  temperature-coefficient  in  ordinary  chemical 
reactions.  Peter  has  also  discovered  the  interesting  fact  that  the 
temperature-coefficient  alters  with  age,  usually  but  not  always 
decreasing  as  time  goes  on*: 

Sphaerechinus    Segmentation    Q^q  =2-29    Q^  =  1-09 
Later  stages  2-03  1-07 

Echinus  Segmentation  2-30  1-09 

Later  stages  2-08  1-08 

Rana  Segmentation  2-23  1-08 

Later  stages  3-34  1-13 

Furthermore,  the  temperature-coefficient  varies  with  the  tem- 
perature itself,  falhng  as  the  temperature  rises — a  rule  which  Van't 
Hoff  shewed  to  hold  in  ordinary  chemical  operations.  Thus  in  Rana 
the  temperature-coefficient  (Qiq)  at  low  temperatures  may  be  as 
high  as  5-6;  which  is  just  another  way  of  saying  that  at  low 
temperatures  development  is  exceptionally  retarded. 

As  the  several  stages  of  development  are  accelerated  by  warmth, 
so  is  the  duration  of  each  and  all,  and  of  life  itself,  proportionately 
curtailed.  The  span  of  life  itself  may  have  its  temperature- 
coefficient — in  so  far  as  Life  is  a  chemical  process,  and  Death  a 
chemical  result.  In  hot  climates  puberty  comes  early,  and  old  age 
(at  least  in  women)  follows  soon;  fishes  grow  faster  and  spawn 
earlier  in  the  Mediterranean  than  in  the  North  Sea.  Jacques  Loeb  f 
found  (in  complete  agreement  with  the  general  case)  that  the  larval 
stages  of  a  fly  are  abbreviated  by  rise  of  temperature;  that  the 
mean  duration  of  life  at  various  temperatures  can  be  expressed  by 
a  temperature-coefficient  of  the  usual  order  of  magnitude ;  that  this 
coefficient  tends,  as  usual,  to  fall  as  the  temperature  rises;  and 
lastly — what  is  not  a  little  curious — ^that  the  coefficient  is  very  much 
the  same,  in  fact  all  but  identical,  for  the  larva,  pupa  and  imago  of 
the  fly. 

*  The  diflferences  are,  after  all,  of  small  order  of  magnitude,  as  is  all  the  better 
seen  when  we  reduce  the  ten-degree  to  one-degree  coefficients. 

t  J-  Loeb  and  Northrop,  On  the  influence  of  food  and  temperature  upon  the 
duration  of  life,  Journ.  Biol  Chemistry,  xxxii,  pp.  103-121,  1917. 


Ill 


OF  TEMPERATURE  COEFFICIENTS 


231 


Temperature-coefficients  (Q^q)  of  Drosophila 


Larva 

Pupa 

Imago 

15-20°  C. 

115 

117 

118 

20-25°  C. 

106 

1-08 

107 

And  Japanese  students,  studying  a  little  fresh- water  crustacean,  have 
carried  the  experiment  much  beyond  the  range  of  Van't  Hoff 's  law, 
and  have  found  length  of  hfe  to  rise  rapidly  to  a  maximum  at  about 
13-14°  C,  and  to  fall  slowly,  in  a  skew  curve,  thereafter*  (Fig.  67). 


20  30  40 

Temperature,  °C. 
Fig.  67.     Length  of  life,  at  various  temperatures,  in  a  water-flea. 

If  we  now  summarise  the  various  temperature-coefficients  (Q^) 
which  we  have  happened  to  consider,  we  are  struck  by  their 
remarkably  close  agreement: 

Yeast  Qi  =  M3 


Lupin 

M6 

Maize 

1-20 

Pea 

1-09 

Echinoids 

1-08 

Drosophila  (mean)" 

M2 

Frog,  segmentation 

1-08 

„     tadpole 

M3 

Mean 

M2 

*  A.  Terao  and  T.  Tabaka,  Duration  of  life  in  a  water-flea,  Moina  sp.;   Joum. 
Imp.  Fisheries  Inst.,  Tokyo,  xxv,  No.  3,  March  1930. 


232  THE  RATE  OF  GROWTH  [ch. 

The  constancy  of  these  results  might  tempt  us  to  look  on  the 
phenomenon  as  a  simple  one,  though  we  well  know  it  to  be  highly 
complex.  But  we  had  better  rest  content  to  see,  as  Arrhenius  saw 
in  the  beginning,  a  general  resemblance  rather  than  an  identity 
between  the  temperature-coefficients  in  physico-chemical  and 
biological  processes*. 

It  was  seen  from  the  first  that  to  extend  Van't  Hoff's  law  from  physical 
chemistry  to  physiology  was  a  bold  assumption,  to  all  appearance  largely 
justified,  but  always  subject  to  severe  and  cautious  limitations.  If  it  seemed 
to  simplify  certain  organic  phenomena,  further  study  soon  shewed  how  far 
from  simple  these  phenomena  were.  Living  matter  is  always  heterogeneous, 
and  from  one  phase  to  another  its  reactions  change;  the  temperature- 
coefficient  varies  likewise,  and  indicates  at  the  best  a  summation,  or  integration, 
of  phenomena.  Nevertheless,  attempts  have  been  made  to  go  a  little  further 
towards  a  physical  explanation  of  the  physiological  coefficient.  Van't  Hoff 
suggested  a  viscosity-correction  for  the  temperature -coefficient  even  of  an 
ordinary  chemical  reaction;  the  viscosity  of  protoplasm  varies  in  a  marked 
degree,  inversely  with  the  temperature,  and  the  viscosity-factor  goes,  perhaps, 
a  long  way  to  account  for  the  aberrations  of  the  temperature-coefficient.  It 
has  even  been  suggested  (by  Belehradekf)  that  the  temperature- coefficients 
of  the  biologist  are  merely  those  of  protoplasmic  viscosity.  For  instance,  the 
temperature-coefficients  of  mitotic  cell-division  have  been  shewn  to  alter 
from  one  phase  to  another  of  the  mitotic  process,  being  much  greater  at  the 
start  than  at  the  end| ;  and  so,  precisely,  has  it  been  shewn  that  protoplasmic 
viscosity  is  high  at  the  beginning  and  low  at  the  end  of  the  mitotic  process  §. 

On  seasonal  growth 

There  is  abundant  evidence  in  certain  fishes,  such  as  plaice  and 
haddock,  that  the  ascending  curve  of  growth  is  subject  to  seasonal 
fluctuations  or  interruptions,  the  rate  during  the  winter  months 
bejng  always  slower  than  in  the  months  of  summer.  Thus  the 
Newfoundland  cod  have  their  maximum  growth-rate  in  June,  and 
in  January-February  they  cease  to  grow;  it  is  as  though  we  super- 
imposed a  periodic  annual  sine-curve  upon  the  continuous  curve  of 
growth.  Furthermore,  as  growth  itself  grows  less  and  less  from 
year  to  year,  so  will  the  difference  between  the  summer  and  the 

♦  Cf.  L.  V.  Heilbronn,  Science,  lxii,  p.  268,  1925. 
t  J.  Belehradek,  in  Biol.  Reviews,  v,  pp.  30-58,  1930. 

X  Cf.  E.  Faure-Fremiet,  La  cinetique  du  developpement,  1925;  also  B.  Ephrussi, 
C.R.  cLxxxii,  p.  810,  1926. 

§   See  {int.  al.)  L.  V.  Heilbronn,  The  Colloid  Chemistry  of  Protoplasm,  1928. 


Ill] 


OF  SEASONAL  GROWTH 


233 


winter  rates  grow  less  and  less.  The  fluctuation  in  rate  represents 
a  vibration  which  is  gradually  dying  out;  the  amplitude  of  the 
sine-curve  diminishes  till  it  disappears;  in  short  our  phenomenon 
is  simply  expressed  by  what  is  known  as  a  "damped  sine-curve*." 

Growth  in  height  of  German  military  cadets,  in  half-yearly  periods 

Increment  (em.) 


Height  (cm. 

) 

f 

A 

■> 

Number 
observed 

, 

^ 

Winter 
^-year 

Summer 
^-year 

Age 

October 

April 

October 

Year 

12 

11-12 

139-4 

141-0 

143-3 

1-6 

2-3 

3-9 

80 

12-13 

143-0 

144-5 

1474 

1-5 

2-9 

4-4 

146 

13-14 

147-5 

149-5 

152-5 

2-0 

3-0 

5-0 

162 

14-1.-) 

152-2 

155-0 

158-5 

2-8 

3-5 

6-3 

162 

15-16 

158-5 

160-8 

163-8 

2-3 

30 

5-3 

150 

16-17 

163-5 

165-4 

167-7 

1-9 

2-3 

4-2 

82 

17-18 

167-7 

168-9 

170-4 

1-2 

1-5 

2-7 

22 

18-19 

169-8 

170-6 

171-5 

0-8 

0-9 

1-7 

6 

19-20 

170-7 

171-1 

171-5 
Mean 

0-4 
1-6 

0-4 
2-2 

0-8 

cm.  4 


years 

Fig.  68.    Half-yearly  increments  of  growth,  in  cadets  of  various  ages. 
From  Daffner's  data. 

The  same  thing  occurs  in  man,  though  neither  in  his  case  nor  in 
that  of  the  fish  have  we  sufficient  data  for  its  complete  illustration. 
We  can  demonstrate  the  fact,  however,  by  help  of  certain  measure- 
ments of  the  height  of  German  cadets,  measured  at  half-yearly 
intervals  t-  In  the  accompanying  diagram  (Fig.  68)  the  half-yearly 
increments  are  set  forth  from  the  above  table,  and  it  will  be  seen 


*  The  scales,  on  the  other  hand,  make  most  of  their  growth  during  the  int/er- 
mediate  seasons:  and  with  this  peculiarity,  that  a  few  broad  zones  are  added  to 
the  scale  in  spring,  and  a  larger  number  of  narrow  circuli  in  autumn :  see  Contrib. 
to  Canadian  Biology,  iv,  pp.  289-305,  1929;  Ben  Dawes,  Growth... in  plaice, 
Journ.  M.B.A.  xvii,  pp.  103-174,  1930. 

t  From  Daffner,  Da^  Wachstum  des  Menschen^  p.  329,  1902. 


234 


THE  RATE  OF  GROWTH 


CH. 


that  they  form  two  even  and  entirely  separate  series.  Danish  school- 
boys show  just  the  same  periodicity  of  growth  in  stature. 

The  seasonal  effect  on  visible  growth-rate  is  much  alike  in  fishes 
and  in  man,  in  spite  of  the  fact  that  the  bodily  temperature  of  the  one 
varies  with  the  milieu  externe  and  that  of  the  other  keeps  constant 
to  within  a  fraction  of  a  degree. 

While  temperature  is  the  dominant  cause,  it  is  not  the  only  cause 
of  seasonal  fluctuations  of  growth;  for  alternate  scarcity  and 
abundance  of  food  is  often,  as  in  herbivorous  animals,  the  ostensible 


18 


10 


a 
o 

S 


/   •^A 

-1 1 1 J 1 1 X 

/          /         M 

/             \ 

/        /               '^ 

\                                       I          \ 

/        / 

V                        /        \ 

/         / 
/          / 

\\                /       \ 

y    / 

\\                         /    Steers\ 

/ 

\ 

/ 

\  \                  /                  \ 

v.^ 

y 

\     \                /                      \ 

1 

1         1         1        1 

\              /                HeiVfers 

\ 
\ 
1          1          I.I          t         1         .IN 

JUN  AUG  OCT  DEC  FEB  APR  dUN  AUG  OCT  DEC  FEB  APR 

15  20  25  30  35 

Age  in  months 

Fig.  69.     Seasonal  growth  of  S.  African  cattle:  Sussex  half-breeds. 
After  Schiitte. 


reason.  Before  turnips  came  into  cultivation  in  the  eighteenth 
century  our  own  cattle  starved  for  half  the  year  and  grew  fat  the 
other,  and  in  many  countries  the  same  thing  happens  still.  In 
South  Africa  the  rainy  season  lasts  from  November  to  February; 
by  January  the  grass  is  plentiful,  by  June  or  July  the  veldt  is 
parched  until  rain  comes  again.  Cattle  fatten  from  January  to 
March  or  April;  from  July  to  October  they  put  on  Httle  weight, 
or  lose  weight  rather  than  put  it  on*. 

*  Cf.  D.  J.  Schiitte,  in  Onderstepoort  Journal,  Oct.  1935. 


J 


Ill]  OF  THE  GROWTH  OF  TREES  2^5 

The  growth  of  trees 

Some  sixty  years  ago  Sir  Robert  Christison,  a  learned  and  versatile 
Edinburgh  professor,  was  the  first  to  study  the  "exact  measurement" 
of  the  girth  of  trees*;  and  his  way  of  putting  a  girdle  round  the 
tree,  and  fitting  a  recording  device  to  the  girdle,  is  copied  in  the 
"  dendrographs  "  t  used  in  forestry  today.  The  Edinburgh  beeches 
begin  to  enlarge  their  trunks  in  late  May  or  June,  when  in  full  leaf, 
and  cease  growing  some  three  months  later;'  the  buds  sprout  and 
the  leaves  begin  their  work  before  the  cambium  wakens  to  activity. 
The  beech-trees  in  Maryland  do  likewise,  save  that  the  dates  are 
a  little  earlier  in  the  year;  and  walnut-trees  on  high  ground  in 
Arizona  shew  a  like  short  season  of  growth,  differing  somewhat  in 
date  or  "phase,"  just  as  it  did  in  Edinburgh,  from  one  year  to 
another. 

Deciduous  trees  stop  growing  after  the  fall  of  the  leaf,  but  ever- 
greens grow  all  the  year  round,  more  or  less.  This  broad  fact  is 
illustrated  in  the  following  table,  which  happens  to  relate  to  the 

Mean  monthly  increase  in  girth  of  trees  at  San  Jorge,  Uruguay :  from 
C.  E.  HalVs  data.  Values  given  in  ^percentages  of  total  annual 
increment  J 


Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov.    Dec. 

Evergreens 

91 

8-8 

8-6 

8-9 

7-7 

5-4 

4-3 

6-0 

9-1 

Ill 

10-8  10-2 

Deciduous  trees 

20-3 

14-6 

9-0 

2-3 

0-8 

0-3 

0-7 

13 

3-5 

9-9 

16-7  210 

southern  hemisphere,  and  to  the  climate  of  Uruguay.  The  measure- 
ments taken  were  those  of  the  girth  of  the  tree,  in  mm.,  at  three 
feet  from  the  ground.  The  evergreens  included  Pinus,  Eucalyptus 

*  Sir  R.  Christison,  On  the  exact  measurement  of  trees.  Trans.  Edinb.  Botan.  Soc. 
XIV,  pp.  164-172,  1882.  Cf.  also  Duhamel  du  Monceau,  Des  semis,  et  plantation 
des  arbres,  Paris,  1750.  On  the  general  subject  see  {int.  al.)  Pfeffer's  Physiology 
of  Plants,  II,  Oxford,  1906;  A.  Maliock,  Growth  of  trees,  Proc.  R.S.  (B),  xc, 
pp.  186-191,  1919.  Maliock  used  an  exceedingly  delicate  optical  method,  in 
which  interference-bands,  produced  by  two  contiguous  glass  plates,  shew  a  visible 
displacement  on  the  slightest  angular  movement  of  the  plates,  even  of  the  order 
of  a  millionth  of  an  inch. 

t  W.  S.  Glock,  A.  E.  Douglass  and  G.  A.  Pearson,  Principles ...  of  tree-ring 
analysis,  Carnegie  Inst.  Washington,  No.  486,  1937;  D.  T.  MacDougal,  Tree  Growth, 
Leiden,  1938,  240  pp. 

%  Trans.  Edinb.  Botan.  Soc.  xviii,  p.  456,  1891. 


236 


THE  RATE  OF  GROWTH 


[CH. 


and  Acacia;  the  deciduous  trees  included  Quercus,  Populus,  Robinia 
and  Melia.  The  result  (Fig.  70)  is  much  as  we  might  expect. 
The  deciduous  trees  cease  to  grow  in  winter-time,  and  during 
all  the  months  when  the  trees  are  bare;  during  the  warm  season 
the  monthly  values  are  regularly  graded,  approximately  in  a  sine- 
curve,  with  a  clear  maximum  (in  the  southern  hemisphere)  about 
the  month  of  December.     In  the  evergreens  the  amplitude  of  the 


Fig.  70.     Periodic  annual  fluctuation  in  rate  of  growth  of  trees  in 
the  southern  hemisphere.     From  C.  E.  Hall's  data. 


annual  wave  is  much  less ;  there  is  a  notable  amount  of  growth  all 
the  year  round,  and  while  there  is  a  marked  diminution  in  rate 
during  the  coldest  months,  there  is  a  tendency  towards  equality 
over  a  considerable  part  of  the  warmer  season.  In  short,  the 
evergreens,  at  least  in  this  case,  do  not  grow  the  faster  as  the 
temperature  continues  to  rise;  and  it  seems  probable  that  some  of 
them,  especially  the  pines,  are  definitely  retarded  in  their  growth, 
either  by  a  temperature  above  their  optimum  or  by  a  deficiency  of 
moisture,  during  the  hottest  season  of  the  year. 

Fig.  71  shews  how  a  cypress  never  ceased  to  grow,  but  had  alternate 


Ill 


OF  THE  GROWTH  OF  TREES 


237 


spells  of  quicker  and  slower  growth,  according  to  conditions  of 
which  we  are  not  informed.  Another  figure  (Fig.  72)  illustrates  the 
growth  in  three  successive  seasons  of  the  Calif ornian  redwood,  a  near 
ally  of  the  most  gigantic  of  trees.  Evergreen  though  the  redwood 
is,  its  growth  has  periods  of  abeyance;  there  is  a  second  minimum 
about  midsummer,  and  the  chief  maximum  of  the  year  may  be  that 
before  or  after  this. 


Fig.  71.     Growth  of  cypress  (C.  macrocarpa),  shewing  seasonal  periodicity. 
From  MacDougal's  data:   smoothed  curve. 


1931  1932  1933 

Fig.  72.     Fortnightly  increase  of  girth  in  Californian  redwood  (Sequoia 
sempervirens),  shewing  seasonal  periodicity.   After  MacDougal. 

In  warm  countries  tree-growth  is  apt  to  shew  a  double  maximum, 
for  the  cold  of  winter  and  the  drought  of  summer  are  equally 
antagonistic  tQ  it.  Trees  grow  slower — and  grow  fewer — the  farther 
north  we  go,  till  only  a  few  birches  and  willows  remain, -stunted  and 
old;  it  is  nearly  a  hundred  years  ago  since  Auguste  Bravais* 
shewed  a  steadily  decreasing  growth-rate  in  the  forests  between 
50°  and  70°  N. 

*  Recherches  sur  la  croissance  du  pin  silvestre  dans  le  nord  de  I'Europe,  Mdm. 
couronnees  de  VAcad.  R.  de  Belgique,  xv,  64  pp.,  1840-41. 


238 


THE  RATE  OF  GROWTH 


[CH. 


The  delicate  measuring  apparatus  now  used  shews  sundry  minor 
but  beautiful  phenomena.  A  daily  periodicity  of  growth  is  a 
common  thing*  (Fig.  73).  In  the  tree-cactuses  the  trunk  expands 
by  day  and  shrinks  again  after  nightfall ;  for  the  stomata  close  in  sun- 
light, and  transpiration  is  checked  until  the  sun  goes  down.  But 
it  is  more  usual  for  the  trunk  to  shrink  from  sunrise  until  evening 
and  to  swell  from  sunset  until  dawn ;   for  by  dayhght  the  leaves  lose 


70°  F 


Black  Poplar 

feet  above  ground, 59  inches 


July  24 


26 


Fig.  73.     Growth  of  black  poplar,  shewing  daily  periodicity. 
After  A.  Mallock. 

water  faster,  and  in  the  dark  they  lose  it  slower,  than  the  roots 
replace  it.  The  rapid  midday  loss  of  water  even  at  the  top  of  a 
tail  Sequoia  is  quickly  followed  by  a  measurable  constriction  of  the 
trunk  fifty  or  even  a  hundred  yards  below  f. 


*  The  diurnal  periodicity  is  beautifully  shewn  in  the  case  of  the  hop  by  Johannes 
Schmidt,  C.R.  du  Laboratoire  Carlsberg,  x,  pp.  235-248,  Copenhagen,  1913. 

•f  This  rapid  movement  is  accounted  for  by  Dixon  and  Joly's  "cohesion-theory" 
of  the  ascent  of  sap.  The  leaves  shew  innumerable  minute  menisci,  or  cup-shaped 
water-surfaces,  in  their  intercellular  air-spaces.  As  water  evaporates  from  these 
the  little  cups  deepen,  capillarity  increases  its  pull,  and  suffices  to  put  in  motion 
the  strands  or  columns  of  water  which  run  continuously  through  the  vessels  of 
wood,  and  withstand  rupture  even  under  a  pull  of  100-200  atmospheres.  See 
{int.  al.)  H.  H.  Dixon  and  J.  Joly,  On  the  ascent  of  sap,  Phil.  Trans.  (B),  clxxxvi, 
p.  563,  1895;   also  Dixon's  Transpiration  and  the  Ascent  of  Sap,  8vo,  London,  1914. 


Ill]         OF  THE  SECULAR  GROWTH  OF  TREES        239 

In  the  case  of  trees,  the  seasonal  periodicity  of  growth  and  the 
direct  influence  of  weather  are  both  so  well  marked  that  we  are 
entitled  to  make  use  of  the  phenomenon  in  a  converse  way,  and  to 
draw  deductions  (as  Leonardo  da  Vinci  did*)  as  to  climate  during 
past  years  from  the  varying  rates  of  growth  which  the  tree  has 
recorded  for  us  by  the  thickness  of  its  annual  rings.  Mr  A.  E. 
Douglass,  of  the  University  .of  California,  has  made  a  careful  study 
of  this  question,  and  I  received  from  him  (through  Professor  H.  H. 
Turner)  some  measurements  of  the  average  width  of  the  annual 
rings  in  Californian  redwood,  five  hundred  yfears  old,  in  which  trees 
the  rings  are  very  clearly  shewn.  For  the  first  hundred  years  the 
mean  of  two  trees  was  used,  for  the  next  four  hundred  years  the 
mean  of  five;  and  the  means  of  these  (and  sometimes  of  larger 
numbers)  were  found  to  be  very  concordant.  A  correction  was 
appHed  by  drawing  a  nearly  straight  fine  through  the  curve  for  the 
whole  period,  which  line  was  assumed  to  represent  the  slowly 
diminishing  mean  width  of  annual  ring  accompanying  the  increasing 
size,  or  age,  of  the  tree;  and  the  actual  growth  as  measured  was 
equated  with  this  diminishing  mean.  The  figures  used  give,  then, 
the  ratio  of  the  actual  growth  in  each  year  to  the  mean  growth 
of  the  tree  at  that  epoch. 

It  was  at  once  manifest  that  the  growth-rate  so  determined 
shewed  a  tendency  to  fluctuate  in  a  long  period  of  between  100  and 
200  years.  I  then  smoothed  the  yearly  values  in  groups  of  100 
(by  Gauss's  method  of  "moving  averages"),,  so  that  each  number 
thus  found  represented  the  mean  annual  increase  during  a  century: 
that  is  to  say,  the  value  ascribed  to  the  year  1500  represented  the 
average  annual  growth  during  the  whole  period  between  1450  and 
1550,  and  so  on.  These  values,  so  simply  obtained,  give  us  a  curve 
of  beautiful  and  surprising  smoothness,  from  which  we  draw  the 
direct  conclusion  that  the  climate  of  Arizona,  during  the  last  five 
hundred  years,  has  fluctuated  with  a  regular  periodicity  of  almost 
precisely  150  years.  I  have  drawn,  more  recently,  and  also  from 
Mr  Douglass's  data,  a  similar  curve  for  a  group  of  pine  trees  in 
Calaveras  County  |.     These  trees  are  about  300  years  old,  and  the 

*  Cf.  J.  Playfair  McMurrich,  Leonardo  da  Vinci,  1930,  p.  247. 
t  When  this  was  first  written  I  had  not  seen  Mr  Douglass's  paper  On  a  method 
of  estimating  rainfall  by  the  growth  of  trees.  Bull.  Amer.  Geograph.  Soc.  XLVI, 


240 


THE  RATE  OF  GROWTH 


[CH. 


data  are  reduced,  as  before,  to  moving  averages  of  100  years,  but 
without  further  correction.  The  agreement  between  the  growth- 
rate  of  these  pines  and  that  of  the  great  Sequoias  during  the  same 
period  is  very  remarkable  (Fig.  74). 

We  should  be  left  in  doubt,  so  far  as  these  observations  go, 
whether  the  essential  factor  be  a  fluctuation  of  temperature  or  an 
alternation  of  drought  and  humidity;  but  the  character  of  the 
Arizona  chmate,  and  the  known  facts  of  recent  years,  encourage 
the  behef  that  the  latter  is  the  more  direct  and  more  important 
factor.  In  a  New  England  forest  many  trees  of  many  kinds  were 
studied  after  a  hurricane;  they  shewed  on  the  whole  no  correlation^ 


1440 


CQ 


Fig.  74.  Long-period  fluctuation  in  growth  of  Arizona  redwood  (Sequoia),  from 
A.D.  1390  to  1910;  and  of  yellow  pine  from  Calaveras  County,  from  a.d.  1620 
to  1920.     (Smoothed  in  100-year  periods.) 

between  growth-rate  and  temperature,  with  the  remarkable  exception 
(in  the  conifers)  of  a  clear  correlation  with  the  temperature  of 
March  and  April,  a  month  or  two  before  the  season's  growth  began. 
In  a  cold  spring  the  melting  snows  and  early  rains  ran  off  into  the 
rivers,  in  a  warm  and  early  one  they  sank  into  the  soil  * ;  in  other 
words,  humidity  was  still  the  controlling  factor.  An  ancient  oak' 
tree  in  Tunis  is  said  to  have  recorded  fifty  years  of  abundant  rain. 


pp.  321-335,  1914;  nor,  of  course,  his  great  work  on  Climatic  cycles  and  tree- 
growth,  Carnegie  Inst.  Publications,  1919,  1928,  1936.  Mr  Douglass  does  not 
fail  to  notice  the  long  period  here  described,  but  he  is  more  interested  in  the 
sunspot-cycle  and  other  shorter  cycles  known  to  meteorologists.  See  also  (int.  al.) 
E.  Huntingdon,  The  fluctuating  climate  of  North  America,  Geograph.  Journ. 
Oct.  1912;  and  Otto  Pettersson,  Climatic  variation  in  historic  and  prehistoric 
time,  Svens^M,  Hydrografisk-Biolog.  Skrifter,  v,  1914. 

♦  C.  J.  Lyon,  Amer.  Assoc.  Rep.  1939;   Nature,  Apr.  13,  1940,  p.  595. 


Ill]  OF  THE  INFLUENCE  OF  LIGHT  241 

with  short  intervals  of  drought,  during  the  eighteenth  century ;  then, 
after  1790,  longer  droughts  and  shorter  spells  of  rainy  seasons*. 

It  has  been  often  remarked  that  our  common  European  trees, 
such  as  the  elm  or  the  cherry,  have  larger  leaves  the  farther  north 
we  go ;  but  the  phenomenon  is  due  to  the  longer  hours  of  dayhght 
throughout  the  summer,  rather  than  to  intensity  of  illumination  or 
diJBFerence  of  temperature.  On  the  other  hand,  long  dayhght,  by 
prolonging  vegetative  growth,  retards  flowering  and  fruiting;  and 
late  varieties  of  soya  bean  may  be  forced  into  early  ripeness  by 
artificially  shortening  their  dayhght  at  midsummer  f. 

The  effect  of  ultra-violet  hght,  or  any  other  portion  of  the 
spectrum,  is  part,  and  perhaps  the  chief  part,  of  the  same  problem. 
That  ultra-violet  hght  accelerates  growth  has  been  shewn  both  in 
plants  and  animals f.  In  tomatoes,  growth  is  favoured  by  just  such 
ultra-violet  hght  as  comes  very  near  the  end  of  the  solar  spectrum  §, 
and  as  happens,  also,  to  be  especially  absorbed  by  ordinary  green- 
house glass II .  At  the  other  end  of  the  spectrum,  in  red  or  orange 
light,  the  leaves  become  smaller,  their  petioles  longer,  the  nodes 
more  numerous,  the  very  cells  longer  and  more  attenuated.  It  is 
a  physiological  problem,  and  as  such  it  shews  how  plant-hfe  is 
adapted,  on  the  whole,  to  just  such  rays  as  the  sun  sends;  but  it 
also  shews  the  morphologist  how  the  secondary  effects  of  chmate 
may  so  influence  growth  as  to  modify  both  size  and  form^.  An 
analogous  case  is  the  influence  of  hght,  rather  than  temperature, 
in  modifying  the  coloration  of  organisms,  such  as  certain  butterflies. 

*  Le  chene  Zeem  d'Ain  Draham,  Bull,  du  Directeur  General,  Tunisie,  1927. 

t  That  the  plant  grows  by  turns  in  darkness  and  in  light,  and  has  its  characteristic 
growth-phases  in  each,  longer  >or  shorter  according  to  species  and  variety  and 
normal  habitat,  is  a  subject  now  studied  under  the  name  of  "photoperiodism," 
and  become  of  great  practical  importance  for  the  northerly  extension  of  cereal 
crops  in  Canada  and  Russia.  Cf.  R.  G.  Whyte  and  M.  A.  Oljhovikov,  Nature, 
Feb.  18,  1939. 

I  Cf.  Kuro  Suzuki  and  T.  Hatano,  in  Proc.  Imp.  Acad,  of  Japan,  in,  pp.  94-96, 1927. 
§   Withrow  and  Benedict,  in  Bull,  of  Basic  Scient.  Research,  iii,  pp.  161-174,  1931. 

II  Cf.  E.  C.  Teodoresco,  Croissance  des  plantes  aux  lumieres  de  diverses  longueurs 
d'onde,  ^WTi.  Sc.  Nat,  Bat.  (8),  pp.  141-336,  1929;  N.  Pfeiffer,  Botan.  Gaz.  lxxxv, 
p.  127,  1929;  etc. 

II  See  D.  T.  MacDougal,  Influence  of  light  and  darkness,  etc.,  Mem.  N.  Y.  Botan. 
Garden,  1903,  392  pp.;  Growth  in  trees,  Carnegie  Inst.  1921,  1924,  etc.;  J.  Wiesner, 
Lichtgenuss  der  Pflunzen,  vn,  322  pp.,  1907;  Earl  S.  Johnston,  Smithson.  Misc. 
Contrib.  18  pp.,  1938;  etc.  On  the  curious  effect  of  short  spells  of  light  and  dark- 
ness, see  H.  Dickson,  Proc.  E.S.  (B),  cxv,  pp.  115-123,  1938. 


242  THE  RATE  OF  GROWTH  [ch. 

Now  if  temperature  or  light  affect  the  rate  of  growth  in  strict 
uniformity,  aHke  in  all  parts  and  in  all  directions,  it  will  only  lead 
to  local  races  or  varieties  differing  in  size,  as  the  Siberian  goldfinch 
or  bullfinch  differs  from  our  own.  But  if  there  be  ever  so  Httle  of  a 
discriminating  tendency  such  as  to  enhance  the  growth  of  one  tissue 
or  one  organ  more  than  another*,  then  it  must  soon  lead  to  racial, 
or  even  "specific,"  difference  of  form. 

It  is  hardly  to  be  doubted  that  chmate  has  some  such  dis- 
criminating influence.  The  large  leaves  of  our  northern  trees  are 
an  instance  of  it;  and  we  have  a  better  instance  of  it  still  in  Alpine 
plants,  whose  general  habit  is  dwarfed  though  their  floral  organs 
suffer  little  or  no  reduction  f.  Sunhght  of  itself  would  seem  to  be 
a  hindrance  rather  than  a  stimulant  to  growth;  and  the  familiar 
fact  of  a  plant  turning  towards  the  sun  means  increased  growth  on 
the  shady  side,  or  partial  inhibition  on  the  other. 

More  curious  and  still  more  obscure  is  the  moon's  influence  on 
growth,  as  on  the  growth  and  ripening  of  the  eggs  of  oysters,  sea- 
urchins  and  crabs.  Behef  in  such  lunar  influence  is  as  old  as  Egypt; 
it  is  confirmed  and  justified,  in  certain  cases,  nowadays,  but  the 
way  in  which  the  influence  is  exerted  is  quite  unknown  J. 

Osmotic  factors  in  growth 

The  curves  of  growth  which  we  have  been  studying  have  a 
twofold  interest,  morphological  and  physiological.  To  the  morpho- 
logist,  who  has  learned  to  recognise  form  as  a  "function  of  growth," 
the  most  important  facts  are  these:  (1)  that  rate  of  growth  is  an 
orderly  phenomenon,  with  general  features  common  to  various 
organisms,  each  having  its  own  characteristic  rates,  or  specific 
constants;  (2)  that  rate  of  growth  varies  with  temperature,  and  so 
with  season  and  with  climate,  and  also  with  various  other  physical 
factors,  external  and  internal  to  the  organism ;  (3)  that  it  varies  in 
different  parts  of  the  body,  and  along  various  directions  or  axes: 

*  Or  as  we  might  say  nowadays,  have  a  different  "threshold  value"  in  one 
organ  to  another. 

t  Cf.  for  instance,  NageH's  classical  account  of  the  effect  of  change  of  habitat 
on  alpine  and  other  plants,  Sitzungsber.  Baier.  Akad.  Wiss.  1865,  pp.  228-284. 

X  Cf.  Munro  Fox,  Lunar  periodicity  in  reproduction,  Proc.  R.S.  (B),  xcv, 
pp.  523-550,  1935;   also  Silvio  Ranzi,  Pubblic.  Slaz.  Zool.  Napoli,  xi,  1931. 


Ill]  OF  OSMOTIC  FACTORS  IN  GROWTH  243 

such  variations  being  harmoniously  "graded,"  or  related  to  one 
another  by  a  "principle  of  continuity,"  so  giving  rise  to  the 
characteristic  form  and  dimensions  of  the  organism  and  to  the 
changes  of  form  which  it  exhibits  in  the  course  of  its  development. 
To  the  physiologist  the  phenomenon  of  growth  suggests  many  other 
considerations,  and  especially  the  relation  of  growth  itself  to  chemical 
and  physical  forces  and  energies. 

To  be  content  to  shew  that  a  certain  rate  of  growth  occurs  in 
a  certain  organism  under  certain  conditions,  or  to  speak  of  the 
phenomenon  as  a  "reaction"  of  the  living  organism  to  its  environ- 
ment or  to  certain  stimuh,  would  be  but  an  example  of  that  "lack 
of  particularity"  with  which  we  are  apt  to  be  all  too  easily  satisfied. 
But  in  the  case  of  growth  we  pass  some  little  way  beyond  these 
limitations:  to  this  extent,  that  an  affinity  with  certain  types  of 
chemical  and  physical  reaction  has  been  recognised  by  a  great  number 
of  physiologists*. 

A  large  part  of  the  phenomenon  of  growth,  in  animals  and  still 
more  conspicuously  in  plants,  is  associated  with  "turgor,"  that  is 
to  say,  is  dependent  on  osmotic  conditions.  In  other  words,  the 
rate  of  growth  depends  (as  we  have  already  seen)  as  much  or  more 
on  the  amount  of  water  taken  up  into  the  living  cells  f,  as  on  the 
actual  amount  of  chemical  metabohsm  performed  by  them;  and 
sometimes,  as  in  certain  insect-larvae,  we  can  even  distinguish 
between  tissues  which  grow  by  increase  of  cell-size,  the  result 'of 
imbibition,  and  others  which  grow  by  multiplication  of  their 
constituent  cells  |.    Of  the  chemical  phenomena  which  result  in  the 

*  Cf.  F.  F.  Blackman,  Presidential  Address  in  Botany,  Brit.  Assoc,  Dublin, 
1908.  The  idea  was  first  enunciated  by  Baudrimont  and  St  Ange,  Recherches  sur 
le  developpement  du  foetus,  Mem.  Acad.  Sci.  xi,  p.  469,  1851. 

t  Cf.  J.  Loeb,  Untersuckungen  zur  physiologischen  Morphologie  der  Tiere,  1892; 
also  Experiments  on  cleavage,  Journ.  Morphology,  vii,  p.  253,  1892;  Ueber  die 
Dynamik  des  tierischen  Wachstums,  Arch.  f.  Entw.  Mech.  xv,  p.  669,  1902-3; 
Davenport,  On  the  role  of  water  in  growth,  Boston  Soc.  N.H.  1897;  Ida  H.  Hyde 
in  Amer.  Journ.  Physiology,  xii,  p.  241,  1905;  Bottazzi,  Osmotischer  Druck  und 
elektrische  Leitungsfahigkeit  der  Fliissigkeiten  der  Organismen,  in  Asher-Spiro's 
Ergebnisse  der  Physiologic,  vn,  pp.  160-402,  1908;  H.  A.  Murray  in  Journ.  Gener. 
Physiology,  ix,  p.  1,  1925;  J.  Gray,  The  role  of  water  in  the  evolution  of  the 
terrestrial  vertebrates,  Journ.  Exper.  Biology,  vi,  pp.  26-31,  1928;  and  A.  N.  J.  Heyn, 
Physiology  of  cell-elongation,  Botan.  Review,  vi,  pp.  515-574,  1940. 

X  Cf.  C.  A.  Berger,  Carnegie  Inst,  of  Washington,  Contributions  to  Embryology, 
xxvu,  1938. 


244 


THE  RATE  OF  GROWTH 


[CH. 


actual  increase  of  protoplasm  we  shall  speak  presently,  but  the  role 
of  water  in  growth  deserves  a  passing  word,  even  in  our  morpho- 
logical enquiry. 

The  lower  plants  only  Uve  and  grow  in  abundant  moisture;  lew 
fungi  continue  growing  when  the  humidity  falls  below  85  per  cent, 
of  saturation,  and  the  mould-fungi,  such  as  Penicillium,  need  more 
moisture  still  (Fig.  75).     Their  hmit  is  reached  a  little  below  90%. 


Humidity  ^ 

I.  75.     Growth  of  PeniciUium  in  relation  to  humidity. 

Growth  of  PeniciUium  (at  25°  C.)  * 


Humidity 

Growth  per  hour 

(%  of  saturation) 

(mm.) 

1000 

7-7 

97-0 

5-0 

94-2 

1-0 

926 

0-5 

90-8 

0-3 

Among  th^  coelenterate  animals  growth  and  ultimate  size  depend 
on  Httle  more  than  absorption  of  water  and  consequent  turgescence, 
the  process,  she  wing  itself  in  simple  ways.  A  sea-anemone  may  live 
to  ^n  immense  agef,  but  its  age  and  size  have  Httle  to  do  with  one 


*  From  R.  G.  Tomkins,  Studies  of  the  growth  of  moulds,  Proc.  B.S.  (B),  cv, 
pp.  375-^01,  1929. 

t  Like  Sir  John  Graham  Dalyell's  famous  "Granny,"  and  Miss  Nelson's  family 
of  CereiLS  (not  Sagartia)  of  which  one  still  lives  at  over  80  years  old.  Cf.  J.  H. 
Ashworth  and  Nelson  Annandale,  in  Trans.  R.  Physical  Soc.  Edin.  xxv,  pp.  1-14 
1904. 


Ill]  OF  TURGESCENCE  245 

another.  It  has  an  upper  limit  of  size  vaguely  characteristic  of  the 
species,  and  if  fed  well  and  often  it  may  reach  it  in  a  year ;  on  stinted 
diet  it  grows  slowly  or  may  dwindle  down;  it  may  be  kept  at 
wellnigh  what  size  one  pleases.  Certain  full-grown  anemones  were 
left  untended  in  war-time,  unfed  and  in  water  which  evaporated 
down  to  half  its  bulk;  they  shrank  down  to  little  beads,  and  grew 
up  again  when  fed  and  cared  for. 

Loeb  shewed,  in  certain  zoophytes,  that  not  only  must  the  cells 
be  turgescent  in  order  to  grow,  but  that  this  turgescence  is  possible 
only  so  long  as  the  salt-water  in  which  the  cells  lie  does  not  overstep 
a  certain  limit  of  concentration:  a  limit  reached,  in  the  case  of 
Tubularia,  when  the  salinity  amounts  to  about  3-4  per  cent.  Sea- 
water  contains  some  3-0  to  3-5  per  cent,  of  salts  in  the  open  sea, 
but  the  salinity  falls  much  below  this  normal,  to  about  2-2  per  cent., 
before  Tubularia  exhibits  its  full  turgescence  and  maximal  growth; 
a  further  dilution  is  deleterious  to  the  animal.  It  is  likely  enough 
that  osmotic  conditions  control,  after  this  fashion,  the  distribution 
and  local  abundance  of  many  zoophytes.  Loeb  has  also  shewn* 
that  in  certain  fish-eggs  (e.g.  of  Fundulus)  an  increasing  concentration, 
leading  to  a  lessening  water-content  of  the  egg,  retards  the  rate  of 
segmentation  and  at  last  arrests  it,  though  nuclear  division  goes  on 
for  some  time  longer. 

The  eggs  of  many  insects  absorb  water  in  large  quantities,  even 
doubling  their  weight  thereby,  and  fail  to  develop  if  drought  prevents 
their  doing  so ;  and  sometimes  the  egg  has  a  thin- walled  stalk,  or  else 
a  "hydropyle,"  or  other  structure  by  which  the  water  is  taken  inf. 

In  the  frog,  according  to  Bialaszewicz  J,  the  growth  of  the  embryo 
while  within  the  vitelline  membrane  depends  wholly  on  absorption 
of  water.  The  rate  varies  with  the  temperature,  but  the  amount 
of  water  absorbed  is  constant,  whether  growth  be  fast  or  slow. 
Moreover,  the  successive  changes  of  form  correspond  to  definite 
quantities  of  water  absorbed,  much  of  which  water  is  intracellular. 
The  solid  residue,  as  Davenport  Has  also  shewn,  may  even  diminish 


*  P finger's  Archiv,  lv,  1893. 

t  Cf.  V.  B.  Wigglesworth,  hised  Physiology,  1939,  p.  2. 

%  Beitrage  zur  Kenntniss  d.  Wachstumsvorgange  bei  Amphibienerabryonen,  Bull. 
Acad.  Sci.  de  Cracovie,  1908,  p.  783;  also  A.  Drzwina  and  C.  Bohn,  De  Taction. .  .dea 
solutions  salines  sur  les  larves  des  batraciens,  ibuL  1906. 


246  THE  RATE  OF  GROWTH  [ch. 

notably,  while  all  the  while  the  embryo  continues  to  grow  in  bulk 
and  weight.  But  later  on,  and  especially  in  the  higher  animals,  the 
water-content  diminishes  as  growth  proceeds  and  age  advances; 
and  loss  of  water  is  followed,  or  accompanied,  by  retardation  and 
cessation  of  growth.  A  crab  loses  water  as  each  phase  of  growth 
draws  to  an  end  and  the  corresponding  moult  approaches;  but  it 
absorbs  water  in  large  quantities  as  soon  as  the  new  period  of 
growth  begins*.  Moreover,  that  water  is  lost  as  growth  goes  on  has 
been  shewn  by  Davenport  for  the  frog,  by  Potts  for  the  chick,  and 
particularly  by  Fehhng  in  the  case  of  man.  Fehhng's  results  may 
be  condensed  as  follows : 

Age  in  weeks  (man)         6  17  22  24  26  30  35  39 

Percentage  of  water      97-5       91-8       92-0      89-9       86-4       83-7       82-9       74-2 

I 

The  following  illustrate  Davenport's  results  for  the  frog: 


Age  in  weeks  (frog) 

1 

2 

5 

7 

9 

14 

41 

84 

Percentage  of  water 

56-3 

58-5 

76-7 

89-3 

931 

950 

90-2 

87-5 

The  following  table  epitomises  the  drying-off  of  ripening  maize  f; 
it  shews  how  ripening  and  withering  are  closely  akin,  and  are  but 
two  phases  of  senescence  (Fig.  76): 


Days  (from  August  6) 

0 

22 

35 

49 

56 

63 

Percentage  of  water 

87 

81 

77 

68 

65 

58 

The  bird's  egg  provides  all  the  food  and  all  the  water  which  the 
growing  embryo  needs,  and  to  carry  a  provision  of  water  is  the 
special  purpose  of  the  white  of  the  egg ;  the  water  contained  in  the 
albumen  at  the  beginning  of  incubation  is  just  about  what  the 
chick  contains  at  the  end.  The  yolk  is  not  surrounded  by  water, 
which  would  diffuse  too  quickly  into  it,  nor  by  a  crystalloid  solution, 
whose  osmotic  value  would  soon  increase;  but  by  a  watery  albu- 
minous colloid,  whose  osmotic  pressure  changes  slowly  as  its  charge 
of  water  is  gradually  withdrawn  J. 

*  Cf.  A.  Krogh,  Osmotic  regulation  in  aquatic  animals,  Cambridge,  1939. 
t  Henry  and  Morrison,  1917;  quoted  by  Otto  Glaser,  on  Growth,  time  and  form, 
Biolog.  Reviews,  xiii,  pp.  2-58,  1938. 

X  Cf.  James  Gray,  in  Journ.  Exper.  Biology,  iv,  pp.  214-225,  1926. 


Ill 


OF  GAIN  OR  LOSS  OF  WATER 


247 


Distribution  of  water  in  a  hen's  egg 


Gm. 

of  water  contained  in 

Day  of 
incubation 

^ 

Loss  by 
evaporation 

Gain  by 
combustion 

Albumen 

Embryo 

Yolk 

0 

29-9 

0-0 

8-5 

00 

00 

6 

27-2 

0-4 

8-45 

2-4 

001 

12 

20-4 

4-6 

7-8 

5-6 

0-27 

18 

9-2 

181 

2-3 

8-8 

1-20 

20 

2-2 

27-4 

10 

9-8 

2-00 

The  actual  amount  of  water,  compared  with  the  dry  solids  in  the 
egg,  has  been  determined  as  follows: 


Day  of  incubation  (chick) 

5 

8 

11 

14 

17 

19 

Percentage  of  water 

94-7 

93-8 

92-3 

87-7 

82-8 

82-3 

10 


20 


30         40 
Days 
Fig.  76.     Percentage  of  water  in  ripening  maize.   From  Otto  Glaser. 

We  know  very  httle  of  the  part  which  all  this  water  plays:  how 
much  is  mere  "reaction-medium,"  how  much  is  fixed  in  hydrated 
colloids,  how  much,  in  short,  is  bound  or  unbound.  But  we  see  that 
somehow  or  other  water  is  lost,  and  lost  in  considerable  amount, 
as  the  embryo  draws  towards  completion  and  ceases  for  the  time 
being  to  grow. 

All  vertebrate  animals  contain  much  the  same  amount  of  water 
in  their  living  bodies,  say  85  per  cent,  or  thereby,  however  unequally 


248  THE  RATE  OF  GROWTH  [ch. 

distributed  in  the  tissues  that  water  may  be*.  Land  animals  have 
evolved  from  water  animals  with  Httle  change  in  this  respect, 
though  the  constant  proportion  of  water  is  variously  achieved. 
A.  newt  loses  moisture  by  evaporation  with  the  utmost  freedom,  and 
regains  it  by  no  less  rapid  absorption  through  the  skin;  while  a 
lizard  in  his  scaly  coat  is  less  hable  to  the  one  and  less  capable  of 
the  other,  and  must  drink  to  replace  what  water  it  may  lose. 

We  are  on  the  verge  of  a  difficult  subject  when  we  speak  of  the 
role  of  water  in  the  living  tissues,  in  the  growth  of  the  organism, 
and  in  the  manifold  activities  of  the  cell ;  and.  we  soon  learn,  among 
other  more  or  less  unexpected  things,  that  osmotic  equihbrium  is 
neither  universal  nor  yet  common  in  the  living  organism.  The  yolk 
maintains  a  higher  osmotic  pressure  than  the  white  of  the  egg — so 
long  as  the  egg  is  living;  and  the  watery  body  of  a  jellyfish,  though 
aot  far  off  osmotic  equilibrium,  has  a  somewhat  less  salinity  than 
the  sea-water.  In  other  words,  its  surface  acts  to  some  extent  as 
a  semipermeable  membrane,  and  the  fluid  which  causes  turgescence 
of  the  tissues  is  less  dense  than  the  sea- water  outside!. 

In  most  marine  invertebrates,  however,  the  body-fluids  con- 
stituting the  milieu  interne  are  isotonic  with  the  milieu  externe,  and 
vary  in  these  animals  pari  passu  with  the  large  variations  to  which 
sea- water  itself  is  subject.  On  the  other  hand,  the  dwellers  in 
fresh-water,  whether  invertebrates  or  fishes,  have,  naturally,  a  more 
concentrated  medium  within  than  without.  As  to  fishes,  diiferent 
kinds  shew  remarkable  differences.  Sharks  and  dogfish  have  an 
osmotic  pressure  in  their  blood  and  their  body  fluids  little  diiferent 

*  The  vitreous  humour  is  nearly  all  water,  the  enamel  has  next  to 'none,  the 
grey  matter  has  some  86  per  cent.,  the  bones,  say  22  per  cent. ;  lung  and  kidney 
take  up  mor^  than  they  can  hold,  and  so  become  excretory  or  regulatory  organs. 
Eggs,  whether  of  dogfish,  salmon,  frogs, '  snakes  or  birds,  are  composed,  roughly 
speaking,  of  half  water  and  half  solid  matter. 

t  Cf.  {int.  al.)  G.  Teissier,  Sur  la  teneur  en  eau. .  .de  Chrysaora,  Bull.  Soc.  Biol, 
de  France,  1926,  p.  266.  And  especially  A.  V.  Hill,  R.  A.  Gortner  and  others.  On 
the  state  of  water  in  colloidal  and  living  systems.  Trans.  Faraday  Soc.  xxvi, 
pp.  678-704,  1930.  For  recent  literature  see  (e.g.)  Homer  Smith,  in  Q.  Rev.  Biol. 
vn,  p.  1,  1932;  E.  K.  Marshall,  Physiol.  Rev.  xiv,  p.  133,  1934;  Lovatt  Evans, 
Recent  Advances  in  Physiology,  4th  ed.,  1930;  M.  Duval,  Recherches. .  .sur  le  milieu 
interieur  des  animaux  aquatiques,  Thhe,  Paris,  1925;  Paul  Portier,  Physiologic  des 
animaux  marins.  Chap,  iii,  Paris,  1938;  G.  P.  Wells  and  I.  C.  Ledingham,  Effects 
of  a  hypotonic  environment,  Journ.  Exp.  Biol,  xvii,  pp.  337-352,  1940. 


Ill]  OF  OSMOTIC  REGULATION  249 

from  that  of  the  sea-water  outside:  but  with  certain  chemical 
diiferences,  for  instance  that  the  chlorides  within  are  much 
diminished,  and  the  molecular  concentration  is  eked  out  by  large 
accumulations  of  urea  in  the  blood.  The  marine  teleosts,  on  the 
other  hand,  have  a  much  lower  osmotic  pressure  within  than  that 
of  the  sea-water  outside,  and  only  a  httle  higher  than  that  of  their 
fresh-water  allies.  Some,  hke  the  conger-eel,  maintain  an  all  but 
constant  internal  concentration,  very  different  from  that  outside; 
and  this  fish,  like  others,  is  constantly  absorbing  water  from  the  sea ; 
it  must  be  exuding  or  excreting  salt  continually*.  Other  teleosts 
differ  greatly  in  their  powers  of  regulation  and  of  tolerance,  the 
common  stickleback  (which  we  may  come  across  in  a  pool  or  in  the 
middle  of  the  North  Sea)  being  exceptionally  tolerant  or  "eury- 
halinef."  Physiology  becomes  "comparative"  when  it  deals  with 
differences  such  as  these,  and  Claude  Bernard  foresaw  the  existence 
of  just  such  differences:  "Chez  tous  les  etres  vivants  le  milieu 
interieur,  qui  est  un  produit  de  I'organisme,  conserve  les  rapports 
necessaires  d'echange  avec  le  milieu  exterieur;  mais  a  mesure  que 
I'organisme  devient  plus  parfait  le  milieu  organique  se  specifie,  et 
s'isole  en  quel  que  sorte  de  plus  en  plus  au  milieu  ambiant  J."  Claude 
Bernard  was  building,  if  I  mistake  not,  on  Bichat's  earUer  concept, 
famous  in  its  day,  of  life  as  "une  alternation  habituelle  d'action  de 
la  part  des  corps  exterieurs,*  et  de  reaction  de  la  part  du  corps 
vivant":  out  of  which  grew  his  still  more  famous  aphorism,  "La 
vie  est  I'ensemble  des  fonctions  qui  resistent  a  la  mort§." 

One  crab,  like  one  fish,  differs  widely  from  another  in  its  power 

*  Probably  by  help  of  Henle's  tubules  in  the  kidney,  which  structures  the  dogfish 
does  not  possess.  But  the  gills  have  their  part  to  play  as  water-regulators,  as 
also,  for  instance,  in  the  crab. 

t  The  grey  mullets  go  down  to  the  sea  to  spawn,  but  may  live  and  grow  in 
brackish  or  nearly  fresh- water.  The  several  species  differ  much  in  their  adaptability, 
and  Brunelli  sets  forth,  as  follows,  the  range  of  salinity  which  each  can  tolerate : 

M.  auratus  24-35  per  mille 

saliens  16-40 

chelo  10-40 
capita  5-40 

cephalus  4^0 

X  Introduction  d  Vetude  de  la  medecine  experimentale,  1855,  p.  110.  For  a  dis- 
cussion of  this  famous  concept  see  J.  Barcroft,  "La  fixite  du  milieu  interieur  est 
la  condition  de  la  vie  libre,"  Biol.  Reviews,  viii,  pp.  24-87,  1932. 

§  Sur  la  vie  et  la  mart,  p.  1. 


250  THE  RATE  OF  GROWTH  [ch. 

of  self-regulation;  and  these  physiological  differences  help  to  explain, 
in  both  cases,  the  limitation  of  this  species  or  that  to  more  or  less 
brackish,  or  more  or  less  saline,  waters.  In  deep-sea  crabs  {Hyas, 
for  instance)  the  osnjotic  pressure  of  the  blood  keeps  nearly  to  that 
of  the  milieu  exteme,  and  falls  quickly  and  dangerously  with  any 
dilution  of  the  latter;  but  the  httle  shore-crab  (Cardnus  moenas) 
can  hve  for  many  days  in  sea-water  diluted  down  to  one-quarter  of 
its  usual  sahnity.  Meanwhile  its  own  fluids  dilute  slowly,  but  not 
near  so  far;  in  other  words,  this  crab  combines  great  powers  of 
osmotic  regulation  with ,  a  large  capacity  for  tolerating  osmotic 
gradients  which  are  beyond  its  power  to  regulate.  How  the  unequal 
balance  is  maintained  is  yet  but  httle  understood.  But  we  do  know 
that  certain  organs  or  tissues,  especially  the  gills  and  the  antennary 
gland,  absorb,  retain  or  ehminate  certain  elements,  or  certain  ions, 
faster  than  others,  and  faster  than  mere  diffusion  accounts  for;  in 
other  words,  "ionic*'  regulation  goes  hand  in  hand  with  "osmotic" 
regulation,  as  a  distinct  and  even  more  fundamental  phenomenon*. 
This  at  least  seems  generally  true — and  only  natural — that  quickened 
respiration  and  increased  oxygen-consumption  accompany  all  such 
one-sided  conditions:  in  other  words,  the  "steady  state"  is  only 
maintained  by  the  doing  of  work  and  the  expenditure  of  energyf. 

To  the  dependence  of  growth  on  the  uptake  of  water,  and  to  the 
phenomena  of  osmotic  balance  and  its  regulation,  HoberJ  and  also 
Loeb  were  inclined  to  refer  the  modifications  of  form  which  certain 
phyllopod  Crustacea  undergo  when  the  highly  sahne  waters  which 
they  inhabit  are  further  concentrated,  or  are  abnormally  diluted. 
Their  growth  is  retarded  by  increased  concentration,  so  that 
individuals  from  the  more  saline  waters  appear  stunted  and  dwarfish ; 
and  they  become  altered  or  transformed  in  other  ways,  suggestive 
of  "degeneration,"  or  a  failure  to  attain  full  and  perfect  develop- 

*  See  especially  D.  A.  Webb,  Ionic  regulation  in  Cardnus  moenas,  Proc.  R.S.  (B), 
cxxix,  pp.  107-136,  1940. 

t  In  general  the  fresh- water  Crustacea  have  a  larger  oxygen -consumption  than 
the  marine.  Stenohaline  and  euryhaline  are  terms  applied  nowadays  to  species 
which  are.  confined  to  a  narrow  range  of  salinity,  or  are  tolerant  of  a  wide  one. 
An  extreme  case  of  toleration,  or  adaptability,  is  that  of  the  Chinese  woolly-handed 
crab,  Eriockeir,  which  has  not  only  acclimatised  itself  in  the  North  Sea  but  has 
ascended  the  Elbe  as  far  as  Dresden. 

X  R.  Hober,  Bedeutung  der  Theorie  der  Losungen  fiir  Physiologic  und  Medizin, 
Biol.  Centralbl.  xix,  p.  272,  1899. 


Ill]  OF  OSMOTIC  REGULATION  251 

merit*.  Important  physiological  changes  ensue.  The  consumption 
of  oxygen  increases  greatly  in  the  stronger  brines,  as  more  and  more 
active  "  osmo-regulation  "  is  required.  The  rate  of  multiplication  is 
increased,  and  parthenogenetic  reproduction  is  encouraged.  In  the 
less  sahne  waters  male  individuals,  usually  rare,  become  plentiful, 
and  here  the  females  bring  forth  their  young  alive ;  males  disappear 
altogether  in  the  more  concentrated  brines,  and  then  the  females 
lay  eggs,  which,  however,  only  begin  to  develop  when  the  sahnity 
is  somewhat  reduced. 

The  best-known  case  is  the  little  brine-shrimp,  Artemia  salina, 
found  in  one  form  or  another  all  the  world  over,  and  first  discovered 
nearly  two  hundred  years  ago  in  the  salt-pans  at  Lymington. 
Among  many  allied  forms,  one,  A.  milhausenii,  inhabits  the  natron- 
lakes  of  Egypt  and  Arabia,  where,  under  the  name  of  "loul,"  or 
"Fezzan-worm,"  it  is  eaten  by  the  Arabsf.  This  fact  is  interesting, 
because  it  indicates  (and  investigation  has  apparently  confirmed) 
that  the  tissues  of  the  creature  are  not  impregnated  with  salt,  as 
is  the  medium  in  which  it  hves.  In  short  Artemia,  hke  teleostean 
fishes  in  the  sea,  hves  constantly  in  a  "hypertonic  medium";  the 
fluids  of  the  body,  the  milieu  interne,  are  no  more  salt  than  are  those 
of  any  ordinary  crustacean  or  other  animal,  but  contain  only  some 
0-8  per  cent,  of  NaCl  J ,  while  the  milieu  externe  may  contain  from 
3  to  30  per  cent,  of  this  and  other  salts;  the  skin,  or  body- wall,  of. 
the  creature  acts  as  a  "semi-permeable  membrane,"  through  which 
the  dissolved  salts  are  not  permitted  to  diffuse,  though  water  passes 
freely.  When  brought  into  a  lower  concentration  the  animal  may 
grow  large  and  turgescent,  until  a  statical  equilibrium,  or  steady 
state,  is  at  length  attained. 

Among  the  structural  changes  which  result  from  increased  con- 

*  Schmankewitsch,  Zeitschr.  f.  wiss.  Zool.  xxix,  p.  429,  1877.  Schraankewitsch 
has  made  equally  interesting  observations  on  change  of  size  and  form  in  other 
organisms,  after  some  generations  in  a  milieu  of  altered  density ;  e.g.  in  the  flagellate 
infusorian  Ascinonema  acinus  Biitschli. 

t  These  "Fezzan- worms,"  when  first  described,  were  supposed  to  be  "insects' 
eggs";  cf.  Humboldt^  Personal  Narrative,  vi,  i,  8,  note;  Kirby  and  Spence,  Letter  x. 

X  See  D.  J.  Kuenen,  Notes,  systematic  and  physiological,  on  Artemia,  Arch. 
N4erland.  Zool.  iii,  pp.  365-449,  1939;  cf.  also  Abonyi,  Z.f.  w.  Z.  cxiv,  p.  134,  1915. 
Cf.  Mme.  Medwedewa,  Ueber  den  osmotischen  Druck  der  Haemolymph  v.  Artemia; 
in  Ztsch.  f.  vergl.  Physiolog.  v,  pp.  547-554,  1922. 


252 


THE  RATE  OF  GROWTH 


[CH. 


centration  of  the  brine  (partly  during  the  hfe-time  of  the  individual, 
but  more  markedly  during  the  short  season  which  suffices  for  the 
development  of  three  or  four,  or  perhaps  more,  successive  genera- 
tions), it  is  found  that  the  tail  comes  to  bear  fewer  and  fewer 
bristles,  and  the  tail-fins  themselves  tend  at  last  to  disappear: 
these  changes  corresponding  to  what  have  been  described  as  the 
specific  characters  of  A.  milhausenii,  and  of  a  still  more  extreme 
form,  A.  koppeniana]  while  on  the  other  hand,  progressive  dilution 
of  the  water  tends '  to  precisely  opposite  conditions,  resulting  in 
forms  which  have  also  been  described  as  separate  species,  and  even 


u 


^wwwWWWH 
I 


I 


Artewia  s.str. 


Callaonella 


Fig.  77.  Brine-shrimps  (Artemia),  from  more  or  less  saline  water.  Upper  figures 
shew  tail-segment  and  tail-fins;  lower  figures,  relative  length  of  cephalothorax 
and  abdomen.     After  Abonyi. 

referred  to  a  separate  genus,  Callaonella,  closely  akin  to  Branchipus 
(Fig.  77).  Pari  passu  with  these  changes,  there  is  a  marked  change 
in  the  relative  lengths  of  the  fore  and  hind  portions  of  the  body, 
that  is  to  say,  of  the  cephalothorax  and  abdomen:  the  latter 
growing  relatively  longer,  the  Salter  the  water.  In  other  words, 
not  only  is  the  rate  of  growth  of  the  whole  animal  lessened  by  the 
sahne  concentration,  but  the  specific  rates  of  growth  in  the  parts 
of  its  body  are  relatively  changed.  This  latter  phenomenon  lends 
itself  to  numerical  statement,  and  Abonyi  has  shewn  that  we  may 
construct  a  very  regular  curve,  by  plotting  the  proportionate  length 
of  the  creature's  abdomen  against  the  salinity,  or  density,  of  the 
water;  and  the  several  species  of  Artemia,  with  all  their  other 
correlated  specific  characters,  are  then  found  to  occupy  successive, 
more  or  less  well-defined,  and  more  or  less  extended,  regions  of  the 


Ill] 


OF  THE  BRINE-SHRIMPS 


253 


curve  (Fig.  78).  In  short,  the  density  of  the  water  is  so  clearly 
"specific,"  that  we  might  briefly  define  Artemia  jelskii,  for  instance, 
as  the  Artemia  of  density  1000-1010  (NaCl),  or  all  but  fresh  water, 
and  the  typical  A.  salina  (or  principalis)  as  the  Artemia  of  density 
1018-1025,  and  so  on*. 

Koppeniana 
160  """^ 


140 


120 


100 


1000 


1020 


1080 


1100 


1040  1060 

Density  of  water 
Fig.  78,     Percentage  ratio  of  length  of  abdomen  to  cephalothorax 
in  brine -shrimps,  at  various  salinities.     After  Abonyi. 

These  Artemiae  are  capable  of  living  in  waters  not  only  of  great 
density,  but  of  very  varied  chemical  composition,  and  it  is  hard  to 
say  how  far  they  are  safeguarded  by  semi-permeabihty  or  by  specific 
properties  and  reactions  of  the  living  colloids  "j".     The  natron-lakes, 

*  Different  authorities  have  recognised  from  one  to  twenty  species  of  Artemia. 
Daday  de  Dees  {Ann.  sci.  nat.  1910)  reduces  the  salt-water  forms  to  one  species 
with  four  varieties,  but  keeps  A.  jelskii  in  a  separate  sub-genus.  Kuenen  suggests 
two  species,  A.  salina  and  gracilis,  one  for  the  European  and  one  for  the  American 
forms.  According  to  Schmankewitsch  every  systematic  character  can  be  shewn 
to  vary  with  the  external  medium.  Cf.  Professor  Labbe  on  change  of  characters, 
specific  and  even  generic,  of  Copepods  according  to  the  ^H  of  saline  waters  at 
Le  Croisic,  Nature,  March  10,  1928. 

t  We  may  compare  Wo.  Ostwald's  old  experiments  on  Daphnia,  which  died  in 
a  pure  solution  of  NaCl  isotonic  with  normal  sea-water.  Their  death  was  not  to 
be  explained  on  osmotic  grounds;  but  was  seemingly  due  to  the  fact  that  the 
organic  gels  do  not  retain  their  normal  water- content  save  in  the  presence  of  such 
concentrations  of  MgClj  (and  other  salts)  as  are  present  in  sea-water. 


254  THE  RATE  OF  GROWTH  [ch. 

for  instance,  contain  large  quantities  of  magnesium  sulphate;  and 
the  Artemiae  continue  to  live  equally  well  in  artificial  solutions 
where  this  salt,  or  where  calcium  chloride,  has  largely  replaced  the 
common  salt  of  the  more  usual  habitat.  Moreover,  such  waters  as 
those  of  the  natron-lakes  are  subject  to  great  changes  of  chemical 
composition  as  evaporation  and  concentration  proceed,  owing  to  the 
different  solubilities  of  the  constituent  salts;  but  it  appears  that 
the  forms  which  the  Artemiae  assume,  and  the  changes  which  they 
undergo,  are  identical,  or  indistinguishable,  whichever  of  the  above 
salts  happen  to  exist  or  to  predominate  in  their  saline  habitat.  At 
the  same  time  we  still  lack,  so  far  as  I  know,  the  simple  but  crucial 
experiments  which  shall  tell  us  whether,  in  solutions  of  different 
chemical  composition,  it  is  at  equal  densities,  or  at  isotonic  concen- 
trations (that  is  to  say,  under  conditions  where  the  osmotic  pressure, 
and  consequently  the  rate  of  diffusion,  is  identical),  that  the  same 
changes  of  form  and  structure  are  produced  and  corresponding 
phases  of  equihbrium  attained. 

Sea-water  has  been  described  as  an  instance  of  the  "fitness  of  the 
environment*"  for  the  maintenance  of  protoplasm  in  an  appropriate 
milieu;  but  our  Artemias  suffice  to  shew  how  nature,  when  hard 
put  to  it,  makes  shift  with  an  environment  which  is  wholly  abnormal 
and  anything  but  "fit." 

While  Hober  and  others  f  have  referred  all  these  phenomena  to 
osmosis,  Abonyi  is  inclined  to  believe  that  the  viscosity,  or 
mechanical  resistance,  of  the  fluid  also  reacts  upon  the  organism; 
and  other  possible  modes  of  operation  have  been  suggested.  But 
we  may  take  it  for  certain  that  the  phenomenon  as  a  whole  is  not 
a  simple  one.  We  should  have  to  look  far  in  organic  nature  for 
what  the  physicist  would  call  simple  osmosis  % ;  and  assuredly  there 
is  always  at  work,  besides  the  passive  phenomena  of  intermolecular 


*  L.  H.  Henderson,  The  Fitness  of  the  Environment,  1913. 

t  Cf.  Schmankewitsch,  Z.  f.  w.  Zool.  xxv,  ISTi);  xxix,  1877,.  etc.;  transl.  in 
appendix  to  Packard's  Monogr.  of  N.  American  Phyllopoda,  1^83,  pp.  466-514; 
Daday  de  Dees,  Ann.  Sci.  Nat.  (Zool),  (9),  xi,  1910;  Samter  und  Heymons,  Abh. 
d.  K.  pr.  Akad.  Wiss.  1902;  Bateson,  Mat.  for  the  Study  of  Variation,  1894,  pp. 
96-101;  Anikin,  Mitlh.  Kais.  Univ.  Tomsk,  xiv:  Zool.  Centralhl.  vi,  pp.  756-760, 
1908;  Abonyi,  Z.f.  w.  Zool.  cxiv,  pp.  96-168,  1915  (with  copious  bibliography),  etc. 

%  Cf.  C.  F.  A.  Pantin,  Body  fluids  in  animals,  Biol.  Reviews,  \i,  p.  4,  1931; 
J.  Duclaux,  Chimie  apjMquee  a  la  biologic,  1937,  ii,  chap.  4. 


Ill]  OF  CATALYTIC  ACTION  255 

diffusion,  some  other  activity  to  play  the  part  of  a  regulatory 
mechanism*. 

On  growth  and  catalytic  action 

In  ordinary  chemical  reactions  we  have  to  deal  (1)  with  a  specific 
velocity  proper  to  the  particular  reaction,  (2)  with  variations  due 
to  temperature  and  other  physical  conditions,  (3)  with  variations 
due  to  the  quantities  present  of  the  reacting  substances,  according 
to  Van't  Hoff's  "Law  of  Mass  Action,"  and  (4)  in  certain  cases  with 
variations  due  to  the  presence  of  "catalysing  agents,"  as  BerzeHus 
called  them  a  hundred  years  agof.  In  the  simpler  reactions,  the 
law  of  mass  involves  a  steady  slo wing-down  of  the  process  as  the 
reaction  proceeds  and  as  the  initial  amount  of  substance  diminishes: 
a  phenomenon,  however,  which  is  more  or  less  evaded  in  the  organism, 
part  of  whose  energies  are  devoted  to  the  continual  bringing-up  of 
supphes. 

Catalytic  action  occurs  when  some  substance,  often  in  very 
minute  quantity,  is  present,  and  by  its  presence  produces  or 
accelerates  a  reaction  by  opening  "a  way  round,"  without  the 
catalysing  agent  itself  being  diminished  or  used  up  J.  It  diminishes 
the  resistance  somehow — little  as  we  know  what  resistance  means 

*  According  to  the  empirical  canon  of  physiology,  that,  as  Leon  Fredericq 
expresses  it  (Arch,  rfc  Zool.  1885),  '*L'etre  vivant  est  agence  de  telle  maniere  que 
chaque  influence  pertyrbatrice  provoque  d'elle-meme  la  mise  en  activite  de  Tappareil 
compensateur  qui  doit  neutraliser  et  reparer  le  dommage."  Herbert  Spencer  had 
conceived  a  similar  principle,  and  thought  he  recognised  in  it  the  vis  medicutrix. 
Nahirae.  It  is  the  physiological  analogue  of  the  "principle  of  Le  Chatelier  "  (1888), 
with  this  important  difference  that  the  latter  is  a  rigorous  and  quantitative  law, 
ba8e<i  on  a  definite  and  stable  equilibrium.  The  close  relation  between  the  two  is 
maintained  by  Le  Dantec  {La  titabilite  de  la  Vie,  1910,  p,  24),  and  criticised  by 
Lotka  {Physical  Biology,  p,  283  seq.). 

t  In  a  paper  in  the  Berliner  Jahrbuch  for  1836,  This  paper  was  translated  in 
the  Edinburgh  New  Philosophical  Journal  in  the  following  year;  and  a  curioas 
little  paper  On  the  coagulation  of  albumen,  and  catalysis,  by  Dr  Samuel  Brown, 
followed  in  the  Edinburgh  Academic  Annual  for  1840, 

X  Such  phenomena  come  precisely  under  the  head  of  what  Bacon  called 
Instances  of  Magic:  "By  which  I  mean  those  wherein  the  material  or  efficient 
cause  is  scanty  and  small  as  compared  with  the  work  or  effect  produced;  so  that 
even  when  they  are  common,  they  seem  like  miracles,  some  at  first  sight,  others 
even  after  attentive  consideration.  These  magical  effects  are  brought  about  in 
three  ways. .  .[of  which  one  is]  by  excitation  or  invitaticm  in  another  body,  as  in 
the  magnet  which  excites  numberless  needles  without  losing  any  of  its  virtue,  or 
in  yeast  and  such-like."     Nov.  Org,,  cap.  li. 


256  THE  RATE  OF  GROWTH  [ch. 

in  a  chemical  reaction.  But  the  velocity-curve  is  not  altered  in 
form ;  for  the  amount  of  energy  in  the  system  is  not  affected  by  the 
presence  of  the  catalyst,  the  law  of  mass  exerts  its  eifect,  and  the 
rate  of  action  gradually  slows  down.  In  certain  cases  we  have 
the  remarkable  phenomenon  that  a  body  capable  of  acting  as  a 
catalyser  is  necessarily  formed  as  a  product,  or  by-product,  of  the 
main  reaction,  and  in  such  a  case  as  this  the  reaction- velocity  will 
tend  to  be  steadily  accelerated.  Instead  of  dwindhng  away,  such 
a  reaction  continues  with  an  ever-increasing  velocity:  always 
subject  to  the  reservation  that  limiting  conditions  will  in  time  make 
themselves  felt,  such  as  a  failure  of  some  necessary  ingredient  (the 
"law  of  the  minimum"),  or  the  production  of  some  substance  which 
shall  antagonise  and  finally  destroy  the  original  reaction.  Such  an 
action  as  this  we  have  learned,  from  Ostwald,  to  describe  as  "auto- 
catalysis."  Now  we  know  that  certain  products  of  protoplasmic 
metabohsm — we  call  them  enzymes — are  very  powerful  catalysers, 
a  fact  clearly  understood  by  Claude  Bernard  long  ago*;  and  we 
are  therefore  entitled,  to  that  extent,  to  speak  of  an  autocatalytic 
action  on  the  part  of  protoplasm  itself. 

Going  a  httle  farther  in  the  footsteps  of  Claude  Bernard,  Chodat 
of  Geneva  suggested  (as  we  are  told  by  his  pupil  Monnier)  that 
growth  itself  might  be  looked  on  as  a  catalytic,  or  autocatalytic 
reaction:  "On  peut  bien,  ainsi  que  M.  Chodat  I'a  propose,  considerer 
I'accroissement  comme  une  reaction  chimique  complexe,  dans 
laquelle  le  catalysateur  est  la  cellule  vivante,  et  les  corps  en  presence 
sont  I'eau,  les  sels  et  Facide  carboniquet-" 

A  similar  suggestion  was  made  by  Loeb,  in  connection  with  the 

*  "Les  diastases  contiennent,  en  definitive,  le  secret  de  la  vie.  Or,  les  actions 
diastatiques  nous  apparaissent  comme  des  phenomenes  catalytiques,  en  d'autres 
termes,  des  accelerations  de  vitesse  de  reaction."  Cf.  M.  F.  Porchet,  Rewie 
Scientifique,  18th  Feb.  1911.  For  a  last  word  on  this  subject,  see  W.  Frankenberger, 
Katalytische  Umsetzungen  in  homogenen  u.  enzymatischen  Systemen,  Leipzig,  1937. 

t  Cf.  R.  Chodat,  Principes  de  Botanique  (2nd  ed.),  1907,  p.  133;  A.  Monnier,  La 
loi  d'accroissement  des  vegetaux,  Publ.  de  VInst.  de  Bot.  de  VUniv.  de  Geneve  (7), 
m,  1905.  Cf.  W.  Ostwald,  Vorlesungen  iiber  Naturphilosophie,  1902,  p.  342; 
Wo.  Ostwald,  Zeitliche  Eigenschaften  der  Entwicklungsvorgange,  in  Roux's 
Vortrdge,  Heft  5,  1908;  Robertson,  Normal  growth  of  an  individual,  and  its 
biochemical  significance,  Arch.f.  Entw.  Mech.  xxv,  pp.  581-614;  xxvi,  pp.  108-118, 
1908;  S.  Hatai,  Growth-curves  from  a  dynamical  standpoint,  Anat.  Record,  v, 
p.  373,  1911;  A.  J.  Lotka,  Ztschr.  f.  physikal.  Chemie,  Lxxn,  p.  511,  1910;  lxxx, 
p.  159,  1912;   etc. 


Ill]  OF  AUTOCATALYSIS  257 

synthesis  of  nuclear  protoplasm,  or  nuclein;  for  he  remarked  that, 
as  in  an  autocatalysed  chemical  reaction  the  rate  of  synthesis 
increases  during  the  initial  stage  of  cell-division  in  proportion  to  the 
amount  of  nuclear  matter  already  there.  In  other  words,  one  of 
the  products  of  the  reaction,  i.e.  one  of  the  constituents  of  the 
nucleus,  accelerates  the  production  of  nuclear  from  cytoplasmic 
material.  To  take  one  more  instance,  Blackman  said,  in  the  address 
already  quoted,  that  "the  botanists  (or  the  zoologists)  speak  of 
growth,  attribute  it  to  a  specific  power  of  protoplasm  for  assimila- 
tion, and  leave  it  alone  as  a  fundamental  phenomenon;  but  they 
are  much  concerned  as  to  the  distribution  of  new  growth  in  innu- 
merable specifically  distinct  forms.  While  the  chemist,  on  the 
other  hand,  recognises  it  as  a  famihar  phenomenon,  and  refers  it  to 
the  same  category  as  his  other  known  examples  of  autocatalysis." 

Later  on,  Brailsford  Robertson  upheld  the  autocatalytic  theory 
with  skill  and  learning*;  and  knowing  well  that  growth  was  no 
simple  solitary  chemical  reaction,  he  thought  that  behind  it  lay  some 
one  master-reaction,  essentially  autocatalytic,  by  which  protoplasmic 
synthesis  was  effected  or  controlled.  He  adduced  at  least  one 
curious  case,  in  the  growth  and  multiphcation  of  the  Infusoria, 
which  can  hardly  be  described  otherwise  than  as  catalytic.  Two 
minute  individuals  (of  Enchelys  or  Colpodiuyn)  kept  in  the  same  drop 
of  water,  so  enhance  each  other's  rate  of  asexual  reproduction  that 
it  may  be  many  times  as  great  when  two  are  together  as  when  one 
is  alone;  the  phenomenon  has  been  called  allelocatalysis.  When  a 
single  infusorian  is  isolated,  it  multiplies  the  quicker  the  smaller  the 
drop  it  is  in — a  further  proof  or  indication  that  something  is  being 
given  oif,  in  this  instance  by  the  living  cells,  which  hastens  growth 
and  reproduction.  But  even  the  ordinary  multiplication  of  a 
bacterium,  which  doubles  its  numbers  every  few  minutes  till  (were 
it  not  for  hmiting  factors)  those  numbers  would  be  all  but  incal- 
culable in  a  day,  looks  like  and  has  been  cited  as  a  simple  but  most 
striking  instance  of  the  potentiahties  of  protoplasmic  catalysis. 

It  is  not  necessary  for  us  to  pursue  this  subject  much  further. 

*  T.  B.  Robertson,  The  Chemical  Basis  of  Growth  and  Senescence,  1923;  and 
earlier  papers.  Cf.  his  Multiplication  of  isolated  infusoria,  Biochem.  Journ.  xv, 
pp.  598-611,  1921;  cf.  Journ.  Physiol,  lvi,  pp.  404-412,  1921;  R.  A.  Peters, 
Substances  needed  for  the  growth  of. .  .Colpodiy,m,  Journ.  Physiol,  lv,  p.  1,  1921. 


258  THE  RATE  OE  GROWTH  [ch. 

It  is  sufficiently  obvious  that  the  normal  S-shaped  curve  of  growth 
of  an  organism  resembles  in  its  general  features  the  velocity-curve 
of  chemical  autocatalysis,  and  many  writers  have  enlarged  on  the 
resemblance;  but  the  S-shaped  curve  of  growth  of  a  population 
resembles  it  just  as  well.  When  the  same  curve  depicts  the  growth 
of  an  individual,  and  of  a  population,  and  the  velocity  of  a  chemical 
reaction,  it  is  enough  to  shew  that  the  analogy  between  these  is  a 
mathematical  and  not  a  physico-chemical  one.  The  sigmoid  curve 
of  growth,  common  to  them  all,  is  sufficiently  explained  as  an 
interference  effect,  due  to  opposing  factors  such  as  we  may  use  a 
differential  equation  to  express :  a  phase  of  acceleration  is  followed 
by  a  phase  of  retardation,  and  the  causes  of  both  are  in  each  case 
complex,  uncertain  or  unknown. .  Nor  are  points  of  difference  lacking 
between  the  chemical  and  the  biological  phenomena.  As  the 
chemical  reaction  draws  to  a  close,  it  is  by  the  gradual  attainment 
of  chemical  equihbrium;  but  when  organic  growth  comes  to  an  end, 
it  is  (in  all  but  the  lowest  organisms)  by  reason  of  a  very  different 
kind  of  equilibrium,  due  in  the  main  to  the  gradual  differentiation 
of  the  organism  into  parts,  among  whose  pecuHar  properties  or 
functions  that  of  growth  or  multiphcation  falls  into  abeyance. 

The  analogy  between  organic  growth  and  chemical  autocatalysis 
is  close  enough  to  let  us  use,  or  try  to  use,  just  such  mathematics  as 
the  chemist  applies  to  his  reactions,  and  so  to  reduce  certain  curves 
of  growth  to  logarithmic  formulae.  This  has  been  done  by  many,  and 
with  no  httle  success  in  simple  cases.  So  have  we  done,  partially, 
in  the  case  of  yeast ;  so  the  statisticians  and  actuaries  do  with  human 
populations;  so  we  may  do  again,  borrowing  (for  illustration)  a 
certain  well-known  study  of  the  growing  sunflower  (Figs.  79,  80). 
Taking  our  mathematics  from  elementary  physical  chemistry,  we 
learn  that : 

The  velocity  of  a  reaction  depends  on  the  concentration  a  of 
the  substance  acted  on:  V  varies  as  a, 

V  =  Ka. 

The  concentration  continually  decreases,  so  that  at  time  t  (in  a 
monomolecular  reaction), 


in 


OF  AUTOCATALYSIS 


259 


4         5         6         7 
Time,  in  weeks 


10        II 


Fig.  79.     Growth  of  sunflower-stem :   observed  and  calculated  curves. 
From  Reed  and  Holland. 


cms. 

250 


200 


150 


100 


50 


■  -■   ■               .    * 

•  ^ — 

•  >^ 

/ 

/Calculauted  (autocaialytic) 

/                  curve 

•  / 

•/ 

y^\    1    1    1 

1        1        1        1        1        1        t        1 

Weeks  1 


10       li       12 


Fig.  80.     Growth  of  sunflower-stem :   calculated  (autocatalytic)  curve. 
After  Reed  and  Holland. 


260  THE  RATE  OF  GROWTH  [ch. 

But  if  the  substance  produced  exercise  a  catalytic  effect,  then  the 
velocity  will  vary  not  only  as  above  but  will  also  increase  as  x 
increases:   the  equation  becomes 

V  =  -T-  ^k'x(a  —  x), 

Cut 

which  is  the  elementary  equation  of  autoca.talysis.     Integrating, 

at       a  —  X 

In  our  growth-problem  it  is  sometimes  found  convenient  to  choose 
for  our  epoch,  t',  the  time  when  growth  is  half-completed,  as  the 
chemist  takes  the  time  at  which  his  reaction  is  half-way  through; 
and  we  may  then  write  (with  a  changed  constant) 


This  is  the  physico-chemical  formula  which  Reed  and  Holland 
apply  to  the  growing  sunflower-stem — a  simple  case*.  For  a  we 
take  the  maximum  height  attained,  viz.  254-5  cm. ;  for  t\  the  epoch 
when  one-half  of  that  height  was  reached,  viz.  (by  interpolation) 
about  34-2  days.  Taking  an  observation  at  random,  say  that  for 
the  56th  day,  when  the  stem  was  228-3  cm.  high,  we  have 

K  in  this  case  is  found  to  be  0-043,  and  the  mean  of  all  such 
determinations  t  is  not  far  difierent. 

Applying  this  formula  to  successive  epochs,  we  get  a  calculated 
curve  in  close  agreement  with  the  observed  one;  and  by  well- 
known  statistical  methods  we  confirm,  and  measure,  its  "closeness 
of  fit."  But  jtist  as  the  chemist  must  vary  and  develop  his  funda- 
mental formula  to  suit  the  course  of  more  and  more  comphcated 
reactions,  so  the  biologist  finds  that  only  the  simplest  of  his  curves 

*  H.  S.  Reed  and  R.  H.  Holland,  The  growth-rate  of  an  annual  plant,  Helianthus, 
Proc.  Nat.  Acad,  of  Sci.  (Washington),  v,  p.  135,  1919;  cf.  Lotka,  op.  cit.,  p.  74, 
A  sifnilar  case  is  that  of  a  gourd,  recorded  by  A.  P.  Anderson,  Bull.  Survey, 
Minnesota,  1895,  and  analysed  by  T.  B.  Robertson,  ibid.  pp.  72-75. 

t  Better  determined,  especially  in  more  complex  cases,  by  the  method  of  least 
squares. 


Ill]  ITS  CHEMICAL  ASPECT  261 

of  growth,  or  only  portions  of  the  rest,  can  be  fitted  to  this  simplest 
of  formulae.  In  a  Hfe-time  are  many  ages;  and  no  all-embracing 
formula  covers  the  infant  in  the  womb,  the  suckling  child,  the 
growing  schoolboy,  the  old  man  when  his  work  is  done.  Besides, 
we  need  such  a  formula  as  a  biologist  can  understand !  One  which 
gives  a  mere  coincidence  of  numbers  may  be  of  little  use  or  none, 
unless  it  go  some  way  to  depict  and  explain  the  modus  operandi  of 
growth.  As  d'Ancona  puts  it :  "II  importe  d'apphquer  des  formules 
qui  correspondent  non  seulement  au  point  de  vue  geometrique,  mais 
soient  representees  par  des  valeurs  de  signification  biologique." 
A  mere  curve-diagram  is  better  than  an  empirical  formula;  for  it 
gives  us  at  least  a  picture  of  the  phenomenon,  and  a  qualitative 
answer  to  the  problem. 

Growth  of  sunflower-stem.     (After  Reed  and  Holland) 


1st  diff. 

15-8 

24-4 

33-3 

39-2 

38-4 

31-6 

22-6 

14-4 

8-5 

4-9 

2-8 


The  chemical  aspect  of  growth 

As  soon  as  we  touch  on  such  matters  as  the  chemical  phenomenon 
of  catalysis  we  are  on  the  threshold  of  a  subject  which,  if  we  were 
able  to  pursue  it,  would  lead  us  far  into  the  special  domain  of 
physiology ;  and  there  it  would  be  necessary  to  follow  it  if  we  were 
dealing  with  growth  as  a  phenomenon  in  itself,  instead  of  mainly 
as  a  help  to  our  study  and  comprehension  of  form.  The  whole 
question  of  diet,  of  overfeeding  and  underfeeding*,  would  present 

*  For  example,  A.  S.  Parker  has  shewn  that  mice  suckled  by  rats,  and  conse- 
quently much  overfed,  grow  so  quickly  that  in  three  weeks  they  reach  double  their 
normal  weight;  but  their  development  is  not  accelerated;  Ann.  Appl.  Biol,  xvi, 
1929. 


Height  (( 

cm.) 

j^ 

>  (days) 

Observed 

Calculated 

7 

17-9 

21-9 

14 

34-4 

37-7 

21 

67-8 

62-1 

28 

981 

95-4 

35 

1310 

134-6 

42 

1690 

173-0 

49 

205-5 

204-6 

56 

228-3 

227-2 

63 

247-1 

241-6 

70 

250-5 

250-1 

77 

253-8 

255-0 

84 

254-5 

257-8 

262  THE  RATE  OF  GROWTH  [ch. 

itself  for  discussion*.  But  without  opening  up  this  large  subject, 
we  may  say  one  more  passing  word  on  the  remarkable  fact  that 
certain  chemical  substances,  or  certain  physiological  secretions, 
have  the  power  of  accelerating  or  of  retarding  or  in  some  way 
regulating  growth,  and  of  so  influencing  the  morphological  features 
of  the  organism. 

To  begin  with  there  are  numerous  elements,  such  as  boron, 
manganese,  cobalt,  arsenic,  which  serve  to  stimulate  growth,  or 
whose  complete  absence  impairs  or  hampers  it;  just  as  there  are 
a  few  others,  such  as  selenium,  whose  presence  in  the  minutest 
quantity  is  injurious  or  pernicious.  The  chemistry  of  the  hving 
body  is  more  complex  than  we  were  wont  to  suppose. 

Lecithin  was  shewn  long  ago  to  have  a  remarkable  power  of 
stimulating  growth  in  animals t,  and  accelerators  of  plant-growth, 
foretold  by  Sachs,  were  demonstrated  by  Bottomley  and  others  J; 
the  several  vitamins  are  either  accelerators  of  growth,  or  are  indis- 
pensable in  order  that  it  may  proceed. 

In  the  little  duckweed  of  our  ponds  and  ditches  [Lemna  minor)  the  botanists 
have  found  a  plant  in  which  growth  and  multiplication  are  reduced  to  very 
simple  terms.  For  it  multiplies  by  budding,  grows  a  rootlet  and  two  or  three 
leaves,  and  buds  again;  it  is  all  young  tissue,  it  carries  no  dead  load;  while 
the  sun  shines  it  has  no  lack  of  nourishment,  and  may  spread  to  the  limits  of 
the  pond.  In  one  of  Bottomley's  early  experiments,  duckweed  was  grown 
(1)  in  a  "culture  solution"  without  stint  of  space  or  food,  and  (2)  in  the  same, 
with  the  addition  of  a  little  bacterised  peat  or  "auximone."  In  both  cases  the 
little  plant  spread  freely,  as  in  the  first,  or  Malthusian,  phase  of  a  population 
curve;  but  the  peat  greatly  accelerated  the  rate,  which  was  not  slow  before. 
Without  the  auximone  the  population  doubled  in  nine  or  ten  days,  and  with 
it  in  five  or  six;  but  in  two  months  the  one  was  seventy-fold  the  other ! 

The  subject  has  grown  big  from  small  beginnings.  We  know 
certain  substances,  haematin  being  one,  which  stimulate  the  growth 
of  bacteria,  and  seem  to  act  on  them  as  true  catalysts.  An  obscure 
but  complex  body  known  as  "bios"  powerfully  stimulates  the 
growth  of  yeast;  and  the  so-called  auxins,  a  name  which  covers 
numerous  bodies  both  nitrogenous  and  non-nitrogenous,  serve  in 

*  For  a  brief  resume  of  this  subject  see  Morgan's  Experimental  Zoology,  chap.  xvi. 

t  Hatai,  Amer.  Joum.  Physiology,  x,  p.  57,  1904;  Danilewsky,  C.R.  cxxi,  cxxn, 
1895-96. 

X  W.  B.  Bottomley,  Proc.  R.S.  (B),  lxxxviii,  pp.  237-247,  1914,  and  other 
papers.     O.  Haberlandt,  Beitr.  z.  allgem.  Botanik,  1921. 


Ill]  OF  HORMONES  263 

minute  doses  to  accelerate  the  growth  of  the  higher  plants*.  Some 
of  these  "growth-substances"  have  been  extracted  from  moulds  or 
from  bacteria,  and  one  remarkable  one,  to  which  the  name  auxin 
is  especially  applied,  from  seedhng  oats.  This  last  is  no  enzyme 
but  a  stable  non-nitrogfenous  substance,  which  seems  to  act  by 
softening  the  cell- wall  and  so  facihtating  the  expansion  of  the  cell. 
Lastly  the  remarkable  discovery  has  been  made  that  certain  indol- 
compounds,  comparatively  simple  bodies,  act  to  all  intents  and 
purposes  in  the  same  way  as  the  growth-hormones  or  natural 
auxins,  and  one  of  these  "hetero-auxins."  an  indol-acetic  acidf, 
is  already  in  common  and  successful  use  to  promote  the  growth  and 
rooting  of  cuttings. 

Growth  of  duckweed,  with  and  without  peat-auodmone 

Without  With 


^ 

^ 

eeks              Obs. 

Calc. 

Obs. 

Calc. 

0                     20 

20 

20 

20 

1                      30 

33 

38 

55 

2                     52 

54 

102 

153 

.1                      77 

88 

326 

424 

4                    135 

155 

1,100 

1,173 

5                   211 

237 

3,064 

3,250 

6                   326 

390 

6,723 

8,980 

7                    550 

640 

19,763 

2,490 

8                   1052 

1048 

69,350 

68,800 

Percentage  increase, 

164  o/o 

2770/0 

per  week 

There  are  kindred  matters  not  less  interesting  to  the  morphologist. 
It  has  long  been  known  that  the  pituitary  body  produces,  in  its 
anterior  lobe,  a  substance  by  which  growth  is  increased  and  regulated. 
This  is  what  we  now  call  a  "hormone" — a  substance  produced  in 
one  organ  or  tissue  and  regulating  the  functions  of  another.  In  this 
case  atrophy  of  the  gland  leaves  the  subject  a  dwarf,  and  its  hyper- 

*  The  older  literature  is  summarised  by  Stark,  Ergebn.  d.  Biologies  n,  1906; 
the  later  by  N.  Nielsen,  Jh.  wiss.  Botan.  Lxxm,  1930;  by  Boyson  Jensen,  Die 
Wuchsstojftheorie,  1935;  by  F.  W.  Went  and  K.  V.  Thimann,  Phytohormones,  New 
York,  1937,  and  by  H.  L.  Pearse,  Plant  hormones  and  their  practical  importance. 
Imp.  Bureau  of  Horticulture,  1939.  Cf.  Went,  Bee.  d.  Trav.  Botan.  Neerl.  xxv,  p.  1, 
1928;  A.  N.  J.  Heyn,  ibid,  xxviii,  p.  113,  1931. 

t  Discovered  by  Kogl  and  Kostermans,  Ztschr.  f.  physiol.  Chem.  ccxxxv,  p.  201, 
1934.  Cf.  {int.  al.)  P.  W.  Zimmermann  and  F.  W.  Wilcox  in  Contrib.  Boyce- 
Thompson  Instil.  1935. 


264  THE  RATE  OF  GROWTH  264 

trophy  or  over-activity  goes  to  the  making  of  a  giant;  the  Umb- 
bones  of  the  giant  grow  longer,  their  epiphyses  get  thick  and  clumsy, 
and  the  deformity  known  as  "acromegaly"  ensues*.  This  has 
become  a  famihar  illustration  of  functional  regulation,  by  some 
glandular  or  "endocrinal"  secretion,  some  enzyme  or  harmozone 
as  Gley  called  it,  or  hormone'f  as  Bayliss  and  Starhng  called  it — in 
the  particular  case  where  the  function  to  be  regulated  is  growth, 
with  its  consequent  influence  on  form.  But  we  may  be  sure  that 
this  so-called  regulation  of  growth  is  no  simple  and  no  specific  thing, 
but  imphes  a  far-reaching  and  complicated  influence  on  the  bodily 
metabohsmj. 

Some  say  that  in  large  animals  the  pituitary  is.  apt  to  be  dispro- 
portionately large §;  and  the  giant  dinosaur  Branchiosaurus,  hugest 
of  land  animals,  is  reputed  to  have  the  largest  hypophyseal  recess 
(or  cavity  for  the  pituitary  body)  ever  observed. 

The  thyroid  also  has  its  part  to  play  in  growth,  as  Gudernatsch 
was  the  first  to  shew||;  perhaps  it  acts,  as  Uhlenhorth  suggests, 
by  releasing  the  pituitary  hormone.  In  a  curious  race  of  dwarf 
frogs  both  thyroid  and  pituitary  were  found  to  be  atrophied  ^.  When 
tadpoles  are  fed  on  thyroid  their  legs  grow  out  long  before  the  usual 
time;  on  the  other  hand  removal  of  the  thyroid  delays  metamor- 
phosis, and  the  tadpoles  remain  tadpoles  to  an  unusual  size**. 

The  great  American  bull-frog  (R.  Catesheiana)  fives  for  two  or 
three  years  in  tadpole  form;  but  a  diet  of  thyroid  turns  the  little 
tadpoles  into  bull-frogs  before  they  are  a  month  old  If-    The  converse 

*  Cf.  E.  A.  Schafer,  The  function  ofjthe  pituitary  body,  Proc.  R.S.  (B),  lxxxi, 
p.  442,  1904. 

t  It  is  not  easy  to  dtaw  a  line  between  enzyme  and  vitamin,  or  between  hormone 
and  enzyme. 

X  The -physiological  relations  between  insulin  and  the  pituitary  body  might 
seem  to  indicate  that  it  is  the  carbohydrate  metabolism  which  is  more  especially 
concerned.     Cf.  (e.g.)  Eric  Holmes,  Metabolism  of  the  Living  Tissue,  1937. 

§  Van  der  Horst  finds  this  to  be  the  case  in  Zalophus  and  in  the  ostrich,  compared 
with  smaller  seals  or  birds;   cf.  Ariens  Kappers,  Journ.  Anat.  lxiv,  p.  256,  1930. 

II   Gudernatsch,  in  Arch.  f.  Entw.  Mech.  xxxv,  1912. 

<;;  Eidmann,  ibid,  xlix,  pp.  510-537,  1921. 

**  Allen,  Journ.  Exp.  Zod.  xxiv,  p.  499,  1918.  Cf.  [int.  al.)  E.  Uhlenhuth, 
Experimental  production  of  gigantism,  Journ.  Gen.  Physiol,  ill,  p.  347;  iv,  p.  321, 
1921-22. 

tt  W.  W.  Swingle,  Journ.  Exp.  Zool.  xxiv,  1918;  xxxvn,  1923;  Journ.  Gen. 
Physiol.  I,  II,  1918-19;   etc! 


Ill]  THE  THYROID  GLAND  .       265 

experiment  has  been  performed  on  ordinary  tadpoles*;  with  their 
thyroids  removed  they  remain  normal  to  all  appearance,  but  the 
weeks  go  by  and  metamorphosis  does  not  take  place.  Gill-clefts 
and  tail  persist,  no  limbs  appear,  brain  and  gut  retain  their  larval 
features;  but  months  after,  or  apparently  at  any  time,  the  belated 
tadpoles  respond  to  a  diet  of  thyroid,  and  may  be  turned  into  frogs 
by  means  of  it.  The  Mexican  axolotl  is  a  grown-up  tadpole  which, 
when  the  ponds  dry  up  (as  they  seldom  do),  completes  its  growth 
and  turns  into  a  gill-less,  lung-breathing  newt  or  salamander!;  but 
feed  it  on  thyroid,  even  for  a  single  meal,  and  its  metamorphosis  is 
hastened  and  ensured  {. 

Much  has  been  done  since  these  pioneering  experiments,  all  going 
to  shew  that  the  thyroid  plays  its  active  part  in  the  tissue-changes 
which  accompany  and  constitute  metamorphosis.  It  looks  as  though 
more  thyroid  meant  more  respiratory  activity,  more  oxygen- 
consumption,  more  oxidative  metabohsm,  more  tissue-change,  hence/ 
earlier  bodily  development  §.  Pituitary  and  thyroid  are  very  different 
things;  the  one  enhances  growth,  the  other  retards  it.  Thyroid 
stimulates  metabolism  and  hastens  development,  but  the  tissues 
waste. 

It  is  a  curious  fact,  but  it  has  often  been  observed,  that  starvation 
or  inanition  has,  in  the  long  run,  a  similar  effect  of  hastening 
metamorphosis  II .     The  meaning  of  this  phenomenon  is  unknown. 

An  extremely  remarkable  case  is  that  of  the  "galls",  brought 
into  existence  on  various  plants  in  response  to  the  prick  of  a  small 
insect's  ovipositor.    One  tree,  an  oak  for  instance,  may  bear  galls 

*  Bennett  Allen,  Biol.  Bull,  xxxii,  1917;  Journ.  Exp.  Zool.  xxiv,  1918;  xxx, 
1920;   etc. 

t  Colorado  axolotls  are  much  more  apt  to  metamorphose  than  the  Mexican 
variety. 

J  Babak,  Ueber  die  Beziehung  der  Metamorphose . .  .  zur  inneren  Secretion, 
Centralbl  f.  Physiol,  x,  1913.  Cf.  Abderhalden,  Studien  iiber  die  von  einzelrien 
Organen  hervorgebrachten  Substanzen  mit  spezifischer  Wirkung,  Pfliiger's  Archiv, 
CLXii,  1915. 

§  Certain  experiments  by  M.  Morse  {Journ.  Biol.  Chem.  xix,  1915)  seeihed  to 
shew  that  the  effect  of  thyroid  on  metamorphosis  depended  on  iodine;  l;^t  the 
case  is  by  no  means  clear  (cf.  0.  Shinryo,  Sci.  Rep.  Tohoku  Univ.  iii,  1928,  and 
others).  The  axolotl  is  said  to  shew  little  response  to  experimental  iodine,  and 
its  ally  Necturus  none  at  all  (cf.  B.  M.  Allen,  in  Biol.  Reviews,  xiii,  1939). 

!|  Cf.  Krizensky,  Die  beschleunigende  Ein wirkung  des  Hungerns  auf  die  Meta- 
morphose, Biol.  Centralbl.  xi.iv,  1914.     Cf.  antea,  p.  170. 


266  THE  RATE  OF  GROWTH  [ch. 

of  many  kinds,  well-defined  and  widely  different,  each  caused  to 
grow  out  of  the  tissues  of  the  plant  by  a  chemical  stimulus  contri- 
buted by  the  insect,  in  very  minute  amount;  and  the  insects  are 
so  much  alike  that  the  galls  are  easier  to  distinguish  than  the  flies. 
The  same  insect  may  produce  the  same  gall  on  different  plants, 
for  instance  on  several  species  of  willow ;  or  sometimes  on  different 
parts,  or  tissues,  of  the  same  plant.  Small  pieces  of  a  dead  larva 
have  been  used  to  infect  a  plant,  and  a  gall  of  the  usual  kind  has 
resulted.  Beyerinck  killed  the  eggs  with  a  hot  wire  as  soon  as 
they  were  deposited  in  the  tree,  yet  the  galls  grew  as  usual.  Here, 
as  Needham  has  lately  pointed  out,  is  a  great  field  for  reflection 
and  future  experiment.  The  minute  drop  of  fluid  exuded  by  the 
insect  has  marvellous  properties.  It  is  not  only  a  stimulant  of 
growth,  like  any  ordinary  auxin  or  hormone;  it  causes  the  growth 
of  a  peculiar  tissue,  and  shapes  it  into  a  new  and  specific  form*. 

Among  other  illustrations  (which  are  plentiful)  of  the  subtle 
influence  of  some  substance  upon  growth,  we  have,  for  instance, 
the  growth  of  the  placental  decidua,  which  Loeb  shewed  to  be  due 
to  a  substance  given  off  by  the  corpus  luteum,  lending  to  the  uterine 
tissues  an  enhanced  capacity  for  growth,  to  be  called  into  action  by 
contact  with  the  ovum  or  even  of  a  foreign  body.  Various  sexual 
characters,  such  as  the  plumage,  comb  and  spurs  of  the  cock,  arise 
in  hke  manner  in  response  to  an  internal  secretion  or  "male 
hormone  " ;  and  when  castration  removes  the  source  of  the  secretion, 
well-known  morphological  changes  take  place.  When  a  converse 
change  takes  place  the  female  acquires,  in  greater  or  less  degree, 
characters  which  are  proper  to  the  male:  as  in  those  extreme  cases, 
known  from  time  immemorial,  when  an  old  and  barren  hen  assumes 
the  plumage  of  the  cockf. 

The  mane  of  the  lion,  the  antlers  of  the  stag,  the  tail  of  the  peacock, 
are  all  examples  of  intensifled  differential  growth,  or  localised  and 

*  Joseph  Needham,  Aspects  nouveaux  de  la  chimie  et  de  la  biologic  de  la  croia- 
sance  organisee.  Folia  Morphologica,  Warszawa,  viii,  p.  32,  1938.  On  galls,  see 
{int.  al.)  Cobbold,  Ross  und  Hedicke,  Die  Pflanzengallen,  Jena,  1927;  etc.  And 
on  their  "raorphogenic  stimulus",  cf.  Herbst,  Biolog.  Cblt.,  1894-5,  passim. 

t  The  hen  which  assumed  the  voice  and  plumage  of  the  male  was  a  portent  or 
omen — gallina  cecinit.  The  first  scientific  account  was  John  Hunter's  celebrated 
Account  of  an  extraordinary  pheasant,  and  Of  the  appearance  of  the  change 
of  sex  in  Lady  Tynte's  peahen,  Phil.  Trans,  lxx,  pp.  527,  534,  1780. 


Ill] 


THE  MALE  HORMONES 


267 


sex-linked  hypertrophy;  and  in  the  singular  and  striking  plumage 
of  innumerable  birds  we  may  easily  see  how  enhanced  growth  of  a 
tuft  of  feathers,  perhaps  exaggeration  of  a  single  plume,  is  at 
the  root  of  the  whole  matter.  Among  extreme  instances  we  may 
think  of  the  immensely  long  first  primary  of  the  pennant-winged 
nightjar;  of  the  long  feather  over  the  eye  in  Pteridophora  alberti, 


Fig.  81.     A  single  pair  of  hypertrophied  feathers  in  a  bird-of-paradise, 
Pteridophora  alberti. 


.-I 


Fig.  82.     Unequal  growth  in  the  three  pairs  of  tail-feathers  of  a  humming-bird 
{Loddigesia).     1,  rudimentafy:  2,  short  and  stiff;  3,  long  and  spathulate. 

or  the  Fix  long  plumes  over  or  behind  the  eye  in  the  six-shafted 
bird-of-paradise;  or  among  the  humming-birds,  of  the  long  outer 
rectrix  in  Lesbia,  the  second  outer  one  in  Aethusa,  or  of  the  extra- 
ordinary inequalities  of  the  tail-feathers  of  Loddigesia  mirabilis, 
some  rudimentary,  some  short  and  straight  and  stiff,  and  other  two 
immensely  elongated,  curved  and  spathulate.  The  sexual, hormones 
have  a  potent  influence  on  the  plumage  of  a>bird ;  they  serve,  somehow, 
to  orientate  and  regulate  the  rate  of  growth  from  one  feather-tract 
to  another,  and  from  one  end  to  another,  even  from  one  side  to  the 
other,  of  a  single  feather.     An  extreme  case  is  the  occasional  pheno- 


268  THE  RATE  OF  GROWTH  [ch. 

menon  of  a  "  gynandrous "  feather,  male  and  female  on  two  sides 
of  the  same  vane*. 

While  unequal  or  differential  growth  is  of  pecuhar  interest  to 
the  morphologist,  rate  of  growth  pure  and  simple,  with  all  the 
agencies  which  control  or  accelerate  it,  remains  of  deeper  importance 
to  the  practical  man.  The  live-stock  breeder  keeps  many  desirable 
quahties  in  view:  constitution,  fertility,  yield  and  quahty  of  milk 
or  wool  are  some  of  these;  but  rate  of  growth,  with  its  corollaries 
of  early  maturity  and  large  ultimate  size,  is  generally  more  important 
than  them  all.  The  inheritance  of  size  is  somewhat  complicated, 
and  limited  from  the  breeder's  point  of  view  by  the  mother's 
inability  to  nourish  and  bring  forth  a  crossbred  offspring  of  a  breed 
larger  than  her  own.  A  cart  mare,  covered  by  a  Shetland  sire, 
produces  a  good-sized  foal;  but  the  Shetland  mare,  crossed  with 
a  carthorse,  has  a  foal  a  little  bigger,  but  not  much  bigger,  than 
herself  (Fig.  83).  In  size  and  rate  of  growth,  as  in  other  qualities, 
our  farm  animals  differ  vastly  from  their  wild  progenitors,  or  from 
the  "  un-improved "  stock  in  days  before  Bake  well  and  the  other 
great  breeders  began.  The  improvement  has  been  brought  about 
by  "selection";  but  what  lies  behind?  Endocrine  secretions, 
especially  pituitary,  are  doubtless  at  work;  and  already  the  stock- 
raiser  and  the  biochemist  may  be  found  hand  in  hand. 

If  we  once  admit,  as  we  are  now  bound  to  do,  the  existence  of 
factors  which  by  their  physiological  activity,  and  apart  from  any 
direct  action  of  the  nervous  system,  tend  towards  the  acceleration 
of  growth  and  consequent  modification  of  form,  we  are  led  into  wide 
fields  of  speculation  by  an  easy  and  a  legitimate  pathway.  Professor 
Gley  carries  such  speculations  a  long,  long  way:  for  He  saysf  that 
by  these  chemical  influences  "Toute  une  partie  de  la  construction 
des  etres  parait  s'expliquer  d'une  fayon  toute  mecanique.  La  forteresse, 
si  longtemps  inaccessible,  du  vitahsme  est  entamee.  Car  la  notion 
morphogenique  etait,  suivant  le  mot  de  Dastre  J ,  comme  '  le  dernier 
reduit  de  la  force  vitale'." 

*  See  an  interesting  paper  by  Frank  R,  Lillie  and  Mary  Juhn,  on  The  physiology 
of  development  of  feathers:  I,  Growth-rate  and  pattern  in  the  individual  feather. 
Physiological  Zoology,  v,  pp.  124-184,  1932,  and  many  papers  quoted  therein. 

f  Le  Neo-vitalisme,  Revue  Scientifique,  March  1911. 

X  La  Vie  et  la  Mart,  1902,  p.  43. 


Ill] 


OF  INHERITANCE  OF  SIZE 


269 


The  physiological  speculations  we  need  not  discuss:  but,  to  take 
a  single  example  from  morphology,  we  begin  to  understand  the 
possibihty,  and  to  comprehend  the  probable  meaning,  of  the  all  but 
sudden  appearance  on  the  earth  of  such  exaggerated  and  almost 
monstrous  forms  as  those  of  the  great  secondary  reptiles  and  the 


500 


Pure  Shetland 


10  20 

Age,  in  months 


30 


40 


Fig.  83. 


Effect  of  cross-breeding  on  rate  of  growth  in  Shetland  ponies. 
From  Walton  and  Hammond's  data.* 


great  tertiary  mammals  f.  We  begin  to  see  that  it  is  in  order  to 
account  not  for  the  appearance  but  for  the  disappearance  of  such 
forms  as  these  that  natural  selection  must  be  invoked.  And  we 
then,  I  think,  draw  near  to  the  conclusion  that  what  is  true  of  these 
is  universally  true,  and  that  the  great  function  of  natural  selection 


*  Walton  and  Hammond,  Proc.  R.S.  (B),  No.  840,  p.  317,  1938. 
t  Cf.  also  Dendy,  Evolutionary  Biology,  1912,  p.  408. 


270  THE  RATE  OF  GROWTH  [ch. 

is  not  to  originate*  but  to  remove:  donee  ad  interitum  genus  id 
natura  redegitf. 

The  world  of  things  living,  like  the  world  of  things  inanimate, 
grows  of  itself,  and  pursues  its  ceaseless  course  of  creative  evolution. 
It  has  room,  wide  but  not  unbounded,  for  variety  of  living  form 
and  structure,  as  these  tend  towards  their  seemingly  endless  but 
yet  strictly  hmited  possibilities  of  permutation  and  degree^  it  has 
room  for  the  great,  and  for  the  small,  room  for  the  weak  and  for  the 
strong.  Environment  and  circumstance  do  not  always  make  a 
prison,  wherein  perforce  the  organism  must  either  live  or  die;  for 
the  ways  of  hfe  may  be  changed,  and  many  a  refuge  found,  before 
the  sentence  of  unfitness  is  pronounced  and  the  penalty  of  exter- 
mination paid.  But  there  comes  a  time  when  "variation,"  in  form, 
dimensions,  or  other  qualities  of  the  organism,  goes  further  than  is 
compatible  with  all  the  means  at  hand  of  health  and  welfare  for 
the  individual  and  the  stock;  when,  under  the  active  and  creative 
stimulus  of  forces  from  within  and  from  without,  the  active  and 
creative  energies  of  growth  pass  the  bounds  of  physical  and 
physiological  equilibrium:  and  so  reach  the  Hmits  which,  as  again 
Lucretius  tells  us,  natural  law  has  set  between  what  may  and  what 
may  not  be, 

et  quid  quaeque  queant  per  foedera  natural 
quid  porro  nequeant. 

Then,  at  last,  we  are  entitled  to  use  the  customary  metaphor,  and 
to  see  in  natural  selection  an  inexorable  force  whose  function  is  not 
to  create  but  to  destroy — to  weed,  to  prune,  to  cut  down  and  to 
cast  into  the  fire  J. 

*  So  said  Yves  Delage  {UherediU,  1903,  p.  397):  "La  selection  naturelle  est  un 
principe  admirable  et  parfaitement  juste.  Tout  le  monde  est  d'accord  sur  ce  point. 
Mais  ou  Ton  n'est  pas  d'accord,  c'est  sur  la  limite  de  sa  puissance  et  sur  la  question 
de  savoir  si  elle  pent  engendrer  des  formes  specifiques  nouvelles.  II  semble  bien 
demontre  aujourd'hui  qu'elle  ne  le  pent  pas.'' 

t  Lucret.  v,  875,  "Lucretius  nowhere  seems  to  recognise  the  possibility  of 
improvement  or  change  of  species  by  'natural  selection';  the  animals  remain  as 
they  were  at  the  first,  except  that  the  weaker  and  more  useless  kinds  have  been 
crushed  out.  Hence  he  stands  in  marked  contrast  with  modern  evolutionists." 
Kelsey's  note,  ad  loc. 

X  Even  after  we  have  so  narrowed  its  scope  and  sphere,  natural  selection  is 
still  a  hard  saying ;  for  the  causes  of  extinction  are  wellnigh  as  hard  to  understand 
as  are  those  of  the  origin  of  species.     If  we  assert  (as  has  been  lightly  and  too 


Ill]  OF  REGENERATIVE  GROWTH  271 

Of  regeneration,  or  growth  and  repair 

The  phenomenon  of  regeneration,  or  the  restoration  of  lost  or 
amputated  parts,  is  a  particular  case  of  growth  which  deserves 
separate  consideration.  It  is  a  property  manifested  in  a  high 
degree  among  invertebrates  and  many  cold-blooded  vertebrates, 
diminishing  as  we  ascend  the  scale,  until  it  lessens  down  in  the 
warm-blooded  animals  to  that  vis smedicatrix  which  heals  a  wound. 
Ever  since  the  days  of  Aristotle,  and  still  more  since  the  experiments 
of  Trembley,  Reaumur  and  Spallanzani  in  the  eighteenth  century, 
physiologist  and  psychologist  alike  have  recognised  that  the  pheno- 
menon is  both  perplexing  and  important.  "Its  discovery,"  said 
Spallanzani,  "  was  an  immense  addition  to  the  riches  of  organic  philo- 
sophy, and  an  inexhaustible  source  of  meditation  for  the  philosopher." 
The  general  phenomenon  is  amply  treated  of  elsewhere*,  and  we 
need  only  deal  with  it  in  its  immediate  relation  to  growth. 

Regeneration,  like  growth  in  other  cases,  proceeds  with  a  velocity 
which  varies  according  to  a  definite  law;  the  rate  varies  with  the 
time,  and  we  may  study  it  as  velocity  and  as  acceleration.  Let  us 
take,  as  an  instance,  Miss  M.  L.  Durbin's  measurements  of  the  rate 
.of  regeneration  of  tadpoles'  tails :  the  rate  being  measured  in  terms 
of  length,  or  longitudinal  increment  f.  From  a  number  of  tadpoles, 
whose  average  length  was  in  one  experiment  34  mm.,  and  in  another 
49  mm.,  about  half  the  tail  was  cut  off,  and  the  average  amounts 
regenerated  in  successive  periods  are  shewn  as  follows : 

Days                              3         5         7  10  12  14  17  18  24  28  30 

Amount  regenerated  (mm.): 

First  experiment      1-4      —  3-4  4-3  —  5-2  —  5-5  6-2  —  6o 

Second       „             0-9      2-2  3-7  5-2  60  6-4  7-1  —  7-6  8-2  8-4 

confidently  done)  that  Smilodon  perished  on  account  of  its  gigantic  tusks,  that 
Teleosaurus  was  handicapped  by  its  exaggerated  snout,  or  Stegosaurus  weighed 
down  by  its  intolerable  load  of  armour,  we  may  call  to  mind  kindred  forms  where 
similar  conditions  did  not  lead  to  rapid  extermination,  or  where  extinction  ensued 
apart  from  any  such  apparent  and  visible  disadvantages.  Cf.  F.  A.  Lucas,  On 
momentum  in  variation,  Amer.  Nat.  xli,  p.  46,  1907. 

*  See  Professor  T.  H.  Morgan's  Regeneration  (316  pp.),  1901,  for  a  full  account 
and  copious  bibliography.  The  early  experiments  on  regeneration,  by  V^allisneri, 
Dicquemare,  Spallanzani,  Reaumur,  Trembley,  Baster,  Bonnet  and  others,  are 
epitomised  by  Haller,  Elementa  Physiologiae,  viii,  pp.  156  seq. 

t  Journ.  Exper.  Zool.  vii,  p.  397,  1909. 


272 


THE  RATE  OF  GROWTH 


[CH. 


Both  experiments  give  us  fairly  smooth  curves  of  growth  within 
the  period  of  the  observations;  and,  with  a  shght  and  easy  extra- 
polation, both  curves  draw  to  the  base-hne  at  zero  (Fig.  84).    More- 


Fig.  84.     Curve  of  regenerative  growth  in  tadpoles'  tails. 
From  M.  L.  Durbin's  data. 


Fig.  85.     Tadpoles'  tails : ,  amount  regenerated  daily,  in  mm. 
(Smoothed  curve). 

over,  if  from  the  smoothed  curves  we  deduce  the  daily  increments, 
we  get  (Fig.  85)  a  bell-shaped  curve  similar  to  (or  to  all  appearance 
identical  with)  a  skew  curve  of  error.  In  point  of  fact,  this  instance 
of  regeneration  is   a  very  ordinary  example    of  growth,    with   its 


Ill] 


OF  REGENERATION 


273 


S-shaped  curve  of  integration  and  its  bell-shaped  differential  curve, 
just  as  we  have  seen  it  in  simple  cases,  or  simple  phases,  of  'the 
growth  of  a  population  or  an  individual. 

If  we  amputate  one  limb  of  a  pair  in  some  animal  with  rapid 
powers  of  regeneration,  we  may  compare  from  time  to  time  the 
dimensions  of  the  regenerating  hmb  with  those  of  its  uninjured 
fellow,  and  so  deal  with  a  relative  rather  than  an  absolute  velocity. 
The  legs  of  insect-larvae  are  easily  restored,  but  after  pupation  no 
further  growth  or  regeneration  takes  place.  An  easy  experiment, 
then,  is  to  remove  a  limb  in  larvae  of  various  ages,  and  to  compare 


100        120 


Fig.  86.     Regenerative  growth  in  mealworms'  legs. 

at  leisure  in  the  pupa  the  dimensions  of  the  new  Hmb  with  the  old. 
The  following  much-abbreviated  table  shews  the  gradual  increase 
of  a  regenerating  limb  in  a  mealworm,  up  to  final  equahty  with  the 
normal  Hmb,  the  rate  varying  according  to  the  usual  S-shaped 
curve*  (Fig.  86). 


Rate  of  regeneration  in  the  mealworm  (Tenebrio  moHtor,  larva) 


Days  after  amputation 
%  ratio  of  new  limb  to  old 


0      16      21       25      34      44      58      70      100      121 
0       7       11      20      29      42      71      83       91       100 


*  From  J.  Krizenecky,  Versuch  zur  statisch-graphischen  Untersuchung .  .  .der 
Regenerationsvorgange,  Arch.  f.  Entw.  Mech.  xxxix,  1914;   xlii,  1917. 


274  THE  RATE  OF  GROWTH  [ch. 

Some  writers  have  found  the  curve  of  regenerative  growth  to  be 
different  from  the  curve  of  ordinary  growth,  and  have  commented 
on  the  apparent  difference;  but  they  have  been  misled  (as  it  seems 
to  me)  by  the  fact  that  regeneration  is  seen  from  the  start  or  very 
nearly  so,  while  the  ordinary  curves  of  growth,  as  they  are  usually 
presented  to  us,  date  not  from  the  beginning  of  growth,  but  from 
the  comparatively  late,  and  unimportant,  and  even  fallacious  epoch 
of  birth.  A  complete  curve  of  growth,  starting  from  zero,  has  the 
same  essential  characteristics  as  the  regeneration  curve. 

Indeed  the  more  we  consider  the  phenomenon  of  regeneration, 
the  more  plainly  does  it  shew  itself  to  us  as  but  a  particular  case 
of  the  general  phenomenon  of  growth*,  following  the  same  lines, 
obeying  the  same  laws,  and  merely  started  into  activity  by  the 
special  stimulus,  direct  or  indirect,  caused  by  the  infliction  of  a 
wound.  Neither  more  nor  less  than  in  other  problems  of  physiology 
are  we  called  upon,  in  the  case  of  regeneration,  to  indulge  in 
metaphysical  speculation,  or  to  dwell  upon  the  beneficent  purpose 
which  seemingly  underhes  this  process  of  heahng  and  repair. 

It  is  a  very  general  rule,  though  not  a  universal  one,  that 
regeneration  tends  to  fall  somewhat  short  of  a  complete  restoration 
of  the  lost  part;  a  certain  percentage  only  of  the  lost  tissues  is 
restored.  This  fact  was  well  known  to  some  of  those  old  .investi- 
gators, who,  like  the  Abbe  Trembley  and  hke  Voltaire,  found  a 
fascination  in  the  study  of  artificial  injury  and  the  regeneration 
which  followed  it.  Sir  John  Graham  Dalyell,  for  instance,  says,  in 
the  course  of  an  admirable  paragraph  on  regeneration!:  "The 
reproductive  faculty ...  is  not  confined  to  one  pjortion,  but  may 
extend  over  many;  and  it  may  ensue  even  in  relation  to  the 
regenerated  portion  more  than  once.  Nevertheless,  the  faculty 
gradually  weakens,  so  that  in  general  every  successive  regeneration 
is  smaller  and  more  imperfect  than  the  organisation  preceding  it; 
and  at  length  it  is  exhausted." 

*  The  experiments  of  Loeb  on  the  growth  of  Tubularia  in  various  saline 
solutions,  referred  to  on  p.  24.5,  might  as  well  or  better  have  been  referred  to  under 
the  heading  of  regeneration,  as  they  were  performed  on  cut  pieces  of  the  zoophyte. 
(Cf.  Morgan,  op.  cit.  p.  35.) 

t  Powers  of  the  Creator,  i,  p.  7,  1851.  See  also  Rare  and  Remarkable  Animals, 
u,  pp.  17-19,  90,  1847. 


Ill]  OF  REGENERATION  275 

In  certain  minute  animals,  such  as  the  Infusoria,  in  which  the 
capacity  for  regeneration  is  so  great  that  the  entire  animal  may- 
be restored  from  a  mere  fragment,  it  becomes  of  great  interest  to 
discover  whether  there  be  some  definite  size  at  which  the  fragment 
ceases  to  display  this  power.  This  question  has  been  studied  by 
Lillie*,  who  found  that  in  Stentor,  while  still  smaller  fragments  were 
capable  of  surviving  for  days,  the  smallest  portions  capable  of 
regeneration  were  of  a  size  equal  to  a  sphere  of  about  80 /x  in 
diameter,  that  is  to  say  of  a  volume  equal  to  about  one  twenty- 
seventh  of  the  average  entire  animal.  He  arrives  at  the  remarkable 
conclusion  that  for  this,  and  for  all  other  species  of  animals,  there 
is  a  "minimal  organisation  mass,"  that  is  to  say  a  "minimal  mass 
of  definite  size  consisting  of  nucleus  and  cytoplasm  within  which 
the  organisation  of  the  species  can  just  find  its  latent  expression." 
And  in  like  manner,  Boverif  has  shewn  that  the  fragment  of  a  sea- 
urchin's  egg  capable  of  growing  up  into  a  new  embryo,  and  so 
discharging  the  complete  functions  of  an  entire  and  uninjured  ovum, 
reaches  its  limit  at  about  one-twentieth  of  the  original  egg — other 
writers  having  found  a  Hmit  at  about  one-fourth.  These  magnitudes, 
small  as  they  are,  represent  objects  easily  visible  under  a  low  power 
of  the  microscope,  and  so  stand  in  a  very  different  category  to  the 
minimal  magnitudes  in  which  fife  itself  can  be  manifested,  and 
which  we  have  discussed  in  another  chapter. 

The  Bermuda  "hfe-plant"  (Bryophyllum  calycinum)  has  so 
remarkable  a  power  of  regeneration  that  a  single  leaf,  kept  damp, 
sprouts  into  fresh  leaves  and  rootlets  which  only  need  nourishment 
to  grow  into  a  new  plant.  If  a  stem  bearing  two  opposite  leaves 
be  split  asunder,  the  two  co-equal  sister-leaves  will  produce  (as  we 
might  indeed  expect)  equal  masses  of  shoots  in  equal  times,  whether 
these  shoots  be  many  or  fe^^;  and,  if  one  leaf  of  the  pair  have  part 
cut  off  it  and  the  other  be  left  intact,  the  amount  of  new  growth 

*  F.  R.  Lillie,  The  smallest  parts  of  Stentor  capable  of  regeneration,  Journ. 
Morphology,  xii,  p.  239,  1897. 

t  -Boveri,  Entwicklungsfahigkeit  kernloser  Seeigeleier,  etc..  Arch.  /.  Entw.  Mech. 
u,  1895.  See  also  Morgan,  Studies  of  the  partial  larvae  of  Sphaerechinus,  ibid. 
1895;  J.  Loeb,  On  the  limits  of  divisibility  of  living  matter,  Biol.  Lectures,  1894; 
Pfluger's  Archiv,  lix,  1894,  etc.  Bonnet  studied  the  same  problem  a  hundred 
and  seventy  years  ago,  and  found  that  the  smallest  part  of  the  worm  Lumbriculus 
capable  of  regenerating  was  1^  lines  (3-4  mm.)  long.  For  other  references  and 
discussion  see  H.  Przibram,  Form  und  Formel,  1922,  ch.  v. 


276 


THE  RATE  OF  GROWTH 


[CH. 


will  be  in  direct  and  precise  proportion  to  the  mass  of  the  leaf  from 
which  it  grew.  The  leaf  is  all  the  while  a  living  tissue,  manu- 
facturing material  to  build  its  own  offshoots ;  and  we  have  a  simple 
case  of  the  law  of  mass  action  in  the  relation  between  the  mass  of 
the  leaf  with  its 'included  chlorophyll  and.  that  of  its  regenerated 
oifshoot*. 


16        18      20 
days 

Fig.  87.  Relation  between  the  percentage  amount  of  tail  removed,  the  percentage 
restored,  and  the  time  required  for  its  restoration.  Constructed  from  M.  M. 
Ellis's  data. 

A  number  of  phenomena  connected  with  the  linear  rate  of 
regeneration  are  illustrated  and  epitomised  in  the  accompanying 
diagram  (Fig.  87),  which  I  have  constructed  from  certain  data 
given  by  Ellis  in  a  paper  on  the  relation  of  the  amount  of  tail 
regenerated  to  the  amount  removed,  in  tadpoles.  These  data  are 
summarised  in  the  next  table.     The  tadpoles  were  all  very  much 


*  Jacques  Loeb,  The  law  controlling  the  quantity  and  rate  of  regeneration, 
Proc.  Nat.  Acad.  Sci.  iv,  pp.  117-121,  1918;  Journ.  Gen.  Physiol,  i,  pp.  81-96, 
1918;    Botan.  Oaz.  lxv,  pp.  150-174,  1918. 


Ill]  OF  REGENERATION  277 

of  a  size,  about  40  mm. ;  the  average  length  of  tail  was  very  near 
to  26  mm.,  or  65  per  cent,  of  the  whole  body-length;  and  in  four 
series  of  experiments  about  10,  20,  40  and  60  per  cent,  of  the  tail 
were  severally  removed.  The  amount  regenerated  in  successive 
intervals  of  three  days  is  shewn  in  our  table.  By  plotting  the 
actual  amounts  regenerated  against  these  three-day  intervals  of 
time,  we  may  interpolate  values  for  the  time  taken  to  regenerate 
definite  percentage  amounts,  5  per  cent.,  10  per  cent.,  etc.  of  the 
amount  removed;  and  my  diagram  is  constructed  from  the  four 
sets  of  values  thus  obtained,  that  is  to  say  from  the  four  sets  of 
experiments  which  differed  from  one  another  in  the  amount  of  tail 
amputated.  To  these  we  have  to  add  the  general  result  of  a  fifth 
series  of  experiments,  which  shewed  that  when  as  much  as  75  per 
cent,  of  the  tail  was  cut  off,  no  regeneration  took  place  at  all,  but 
the  animal  presently  died.  In  our  diagram,  then,  each  curve 
indicates  the  time  taken  to  regenerate  n  per  cent,  of  the  amount 
removed.  All  the  curves  converge  towards  infinity  of  time,  when 
the  amount  removed  approaches  75  per  cent,  of  the  whole;  and  all 
start  from  zero,  for  nothing  is  regenerated  where  nothing  had  been 
destroyed. 

The  rate  of  regenerative  growth  in  tadpoles'  tails 
(After  M.  M.  Ellis,  Journ.  Exp.  Zool.  vii,  ^.421,  1909) 


Body 

Tail 

Amount  Per  cent. 

/o 

amount  regenerated 

in  days 

length 

length 

removed 

of  tail 

r 

— ^ 

Series* 

mm. 

mm. 

mm. 

removed 

3 

6 

9 

12 

15 

18 

32 

0 

39-575 

25-895 

3-2 

12-36 

13 

31 

44 

44 

44 

44 

44 

P 

40-21 

26-13 

5-28 

20-20 

10 

29 

40 

44 

44 

44 

44 

R 

39-86 

25-70 

10-4 

40-50 

6 

20 

31 

40 

48 

48 

48 

S 

40-34 

2611 

14-8 

56-7 

0 

16 

33 

39 

45 

48 

48 

*  Each  series  gives  the  mean  of  20  experiments. 

The  amount  regenerated  varies  also  with  the  age  of  the  tadpole, 
and  with  other  factors  such  as  temperature;  in  short,  for  any  given 
age  or  size  of  tadpole,  and  for  various  temperatures,  and  doubtless 
for  other  varying  physical  conditions,  a  similar  diagram  might  be 
constructed!. 

The  power  of  reproducing,  or  regenerating,  a  lost  limb  is  par- 

t  Cf.  also  C.  Zeleny,  Factors  controlling  the  rate  of  regeneration,  Illinois  Biol- 
Monographs,  iii,  p.  1,  1916. 


278  THE   RATE   OF  GROWTH  [ch. 

ticularly  well  developed  in  arthropod  animals,  and  is  sometimes 
accompanied  by  remarkable  modification  of  the  form  of  the 
regenerated  limb.  A  case  in  point,  which  has  attracted  much 
attention,  occurs  in  connection  with  the  claws  of  certain  Crustacea*. 

In  many  of  these  we  have  an  asymmetry  of  the  great  claws, 
one  being  larger  than  the  other  and  also  more  or  less  different  in 
form.  For  instance  in  the  common  lobster,  one  claw,  the  larger 
of  the  two,  is  provided  with  a  few  great  "crushing"  teeth,  while 
the  smaller  claw  has  more  numerous  teeth,  small  and  serrated. 
Though  Aristotle  thought  otherwise,  it  appears  that  the  crushing- 
claw  may  be  on  the  right'  or  left  side,  indifferently ;  whether  it  be 
on  one  or  the  other  is  a  matter  of  "chance."  It  is  otherwise  in 
many  other  Crustacea,  where  the  larger  and  more  powerful  claw  is 
always  left  or  right,  as  the  case  may  be,  according  to  the  species: 
where,  in  other  words,  the  "probability"  of  the  large  or  the  small 
claw  being  left  or  being  right  is  tantamount  to  certainty  f. 

As  we  have  already  seen,  the  one  claw  is  the  larger  because  it 
has  grown  the  faster;  it  has  a  higher  "coefficient  of  growth,"  and 
accordingly,  as  age  advances,  the  disproportion  between  the  two 
claws  becomes  more  and  more  evident.  Moreover,  we  must  assume 
that  the  characteristic  form  of  the  claw  is  a  "function"  of  its 
magnitude ;  the  knobbiness  is  a  phenomenon  coincident  with 
growth,  and  we  never,  under  any  circumstances,  find  the  smaller 
claw  with  big  crushing  teeth  and  the  big  claw  with  Httle  serrate 
ones.  There  are  many  other  somewhat  similar  cases  where  size 
and  form  are  manifestly  correlated,  and  we  have  already  seen,  to 
some  extent,  how  the  phenomenon  of  growth  is  often  accompanied 
by  such  ratios  of  velocity  as  lead  inevitably  to  changes  of  form. 
Meanwhile,  then,  we  must  simply  assume  that  the  essential  difference 
between  the  two  claws  is  one  of  magnitude,  with  which  a  certain 
differentiation  of  form  is  inseparably  associated. 

*  Cf.  H.  Przibram,  Scheerenumkehr  bei  dekapoden  Crustaceen,  Arch.  f.  Entw. 
Mech.  XIX,  pp.  181-247,  1905;  xxv,  pp.  266-344,  1907;  Emmel,  ibid,  xxii,  p.  542, 
1906;  Regeneration  of  lost  parts  in  lobster,  Bep.  Comm.  Inland  Fisheries,  Rhode 
Island,  XXXV,  xxxvi,  1905-6;  Science  (N.S.),  xxvi,  pp.  83-87,  1907;  Zeleny, 
Compensatory  regulation,  Journ.  Exp.  Zool.  ii,  pp.  1-102,  347-369,  1905;   etc. 

t  Lobsters  are  occasionally  found  with  two  symmetrical  claws:  which  are  then 
usually  serrated,  sometimes  (but  very  rarely)  both  blunt-toothed.  Cf.  W.  T.  Caiman, 
P.Z.8.  1906,  pp.  633,  634,  and  reff. 


Ill]  OF  REGENERATION  279 

If  we  amputate  a  claw,  or  if,  as  often  happens,  the  crab  "casts 
it  off,"  it  undergoes  a  process  of  regeneration — it  grows  anew, 
and  does  so  with  an  accelerated  velocity  which  ceases  when 
equilibrium  of  the  parts  is  once  more  attained :  the  accelerated  velocity 
being  a  case  in  point  to  illustrate  that  vis  revulsionis  of  Haller  to 
which  we  have  already  referred. 

With  the  help  of  this  principle,  Przibram  accounts  for  certain 
curious  phenomena  which  accompany  the  process  of  regeneration. 
As  his  experiments  and  those  of  Morgan  shew,  if  the  large  or  knobby 
claw  (A)  be  removed,  there  are  certain  cases,  e.g.  the  common 
lobster,  where  it  is  directly  regenerated.  In  other  cases,  e.g. 
Alpheus*,  the  other  claw  (B)  assumes  the  size  and  form  of  that 
which  was  amputated,  while  the  latter  regenerates  itself  in  the 
form  of  the  lesser  and  weaker  one;  A  and  B  have  apparently 
changed  places.  In  a  third  case,  as  in  the  hermit-crabs,  the  A- 
claw  regenerates  itself  as  a  small  or  5-claw,  but  the  5-claw 
remains  for  a  time  unaltered,  though  slowly  and  in  the  course  of 
repeated  moults  it  later  on  assumes  the  large  and  heavily  toothed 
^-form. 

Much  has  been  written  on  this  phenomenon,  but  in  essence  it  is 
very  simple.  It  depends  upon  the  respective  rates  of  growth,  upon 
a  ratio  between  the  rate  of  regeneration  and  the  rate  of  growth  of 
the  uninjured  limb:  that  is  to  say,  on  the  familiar  phenomenon  of 
unequal  growth,  or,  as  it  has  been  called,  heterogony*.  It  is  com- 
plicated a  little,  however,  by  the  possibility  of  the  uninjured  limb 
growing  all  the  faster  for  a  time  after  the  animal  has  been  relieved 
of  the  other.  From  the  time  of  amputation,  say  of  A,  A  begins  to 
grow  from  zero,  with  a  high  "regenerative"  velocity;  while  B, 
starting  from  a  definite  magnitude,  continues  to  increase  with  its 
normal  or  perhaps  somewhat  accelerated  velocity.  The  ratio 
between  the  two  velocities  of  growth  will  determine  whether,  by  a 
given  time,  A  has  equalled,  outstripped,  or  still  fallen  short  of  the 
magnitude  of  B. 

That  this  is  the  gist  of  the  whole  problem  is  confirmed  (if  con- 
firmation be  necessary)  by  certain  experiments  of  Wilson's.     It  is 

*  E.'  B.  Wilson,  Reversal  of  symmetry  in  Alpheus  heterocheles,  Biol.  Bull,  iv, 
p.  197,  1903. 
t  See  p.  205. 


280  THE   RATE   OF  GROWTH  [ch. 

known  that  by  section  of  the  nerve  to  a  crab's  claw,  its  growth  is 
retarded,  and  as  the  general  growth  of  the  animal  proceeds  the  claw 
comes  to  appear  stunted  or  dwarfed.  Now  in  such  a  case  as  that 
of  Alpheus,  we  have  seen  that  the  rate  of  regenerative  growth  in  an 
amputated  large  claw  fails  to  let  it  reach  or  overtake  the  magnitude 
of  the  growing  little  claw:  which  latter,  in  short,  now  appears  as 
the  big  one.  But  if  at  the  same  time  as  we  amputate  the  big  claw 
we  also  sever  the  nerve  to  the  lesser  one,  we  so  far  slow  down  the 
latter's  growth  that  the  other  is  able  to  make  up  to  it,  and  in  this 
case  the  two  claws  continue  to  grow  at  approximately  equal  rates, 
or  in  other  words  continue  of  coequal  size. 

The  phenomenon  of  regeneration  goes  some  little  way  towards 
helping  us  to  comprehend  the  phenomenon  of  "multiplication  by 
fission,"  as  it  is  exemplified  in  its  simpler  cases  in  many  worms  and 
worm-like  animals.  For  physical  reasons  which  we  shall  have  to 
study  in  another  chapter,  there  is  a  natural  tendency  for  any  tube, 
if  it  have  the  properties  of  a  fluid  or  semi-fluid  substance,  to  break 
up  into  segments  after  it  comes  to  a  certain  length*;  and  nothing 
can  prevent  its  doing  so  except  the  presence  of  some  controlling 
force,  such  for  instance  as  may  be  due  to  the  pressure  of  some 
external  support,  or  some  superficial  thickening  or  other  intrinsic 
rigidity  of  its  own  substance.  If  we  add  to  this  natural  tendency 
towards  fission  of  a  cylindrical  or  tubular  worm,  the  ordinary 
phenomenon  of  regeneration,  we  have  all  that  is  essentially  implied 
in  "reproduction  by  fission."  And  in  so  far  as  the  process  rests 
upon  a  physical  principle,  or  natural  tendency,  we  may  account  for 
its  occurrence  in  a  great  variety  of  animals,  zoologically  dissimilar; 
and  for  its  presence  here  and  absence  there,  in  forms  which  are 
materially  different  in  a  physical  sense,  though  zoologically  speaking 
they  are  very  closely  allied. 

But  the  phenomena  of  regeneration,  like  all  the  other  phenomena 
of  growth,  soon  carry  us  far  afield,  and  we  must  draw  this  long 
discussion  to  a  close. 

*  A  morphological  polarity,  or  essential  difference  between  one  end  and  the  other 
of  a  segment,  is  important  even  in  so  simple  a  case  as  the  internode  of  a  hydroid 
zoophyte;  and  an  electrical  polarity  seems  always  to  accompany  it.  Cf.  A.  P. 
Matthews,  Amer.  Journ.  Physiology,  viii,  j).  294,  1903;  E.  J.  Lund,  Journ.  Exper. 
Zool.  xxxiy,  pp.  477-493;   xxxvi,  pp.  477^94,  1921-22. 


Ill]  THE  RATE  OF  GROWTH  281 

Summary  and  Conclusion 

For  the  main  features  which  appear  to  be  common  to  all  curves 
of  growth  we  may  hope  to  have,  some  day,  a  simple  explanation. 
In  particular  we  should  like  to  know  the  plain  meaning  of  that  point 
of  inflection,  or  abrupt  change  from  an  increasing  to  a  decreasing 
velocity  of  growth,  which  all  our  curves,  and  especially  our  accelera- 
tion curves,  demonstrate  the  existence  of,  provided  only  that  they 
include  the  initial- stages  of  the  whole  phenomenon:  just  as  we 
should  also  hke  to  have  a  full  physical  or  physiological  explanation 
of  the  gradually  diminishing  velocity  of  growth  which  follows,  and 
which  (though  subject  to  temporary  interruption  or  abeyance)  is 
on  the  whole  characteristic  of  growth  in  all  cases  whatsoever.  In 
short,  the  characteristic  form  of  the  curve  of  growth  in  length  (or 
any  other  linear  dimension)  is  a  phenomenon  which  we  are  at 
present  little  able  to  explain,  but  which  presents  us  with  a  definite 
and  attractive  problem  for  future  solution.  It  would  look  as 
though  the  abrupt  change  in  velocity  must  be  due,  either  to  a  change 
in  that  pressure  outwards  from  within  by  which  the  "forces  of 
growth"  make  themselves  manifest,  or  to  a  change  in  the  resistances 
against  which  they  act,  that  is  to  say  the  tension  of  the  surface; 
and  this  latter  force  we  do  not  by  any  means  limit  to  "surface- 
tension"  proper,  but  may  extend  to  the  development  of  a  more  or 
less  resistant  membrane  or  "skin,"  or  even  to  the  resistance  of  fibres 
or  other  histological  elements  binding  the  boundary  layers  to  the 
parts  within*.  I  take  it  that  the  sudden  arrest  of  velocity  is  much 
more  likely  to  be  due  to  a  sudden  increase  of  resistance  than  to  a 
sudden  diminution  of  internal  energies:  in  other  words,  I  suspect 
that  it  is  coincident  with  some  notable  event  of  histological 
differentiation,  such  as  the  rapid  formation  of  a  comparatively  firm 
skin ;  and  that  the  dwindling  of  velocities,  or  the  negative  accelera- 
tion, which  follows,  is  the  resultant  or  composite  effect  of  waning 
forces  of  growth  on  the  one  hand,  and  increasing  superficial  resistance 

*  It  is  natural  to  suppose  the  cell-wall  less  rigid,  or  more  plastic,  in  the  growing 
tissue  than  in  the  full-grown  or  resting  cell.  It  has  been  suggested  that  this  plasticity- 
is  due  to,  or  is  increased  by,  auxins,  whether  in  the  course  of  nature,  or  in  our 
stimulation  of  growth  by  the  use  of  these  bodies.  Cf.  H.  Soding,  Jahrb.  d.  wiss.  Bot. 
Lxxiv,  p.  127;  1931. 


282  THE   RATE   OF   GROWTH  [ch. 

on  the  other.  This  is  as  much  as  to  say  that  growth,  while  its  own 
energy  tends  to  increase,  leads  also,  after  a  while,  to  the  establish- 
ment of  resistances  which  check  its  own  further  increase. 

Our  knowledge  of  the  whole  complex  phenomenon  of  growth  is 
so  scanty  that  it  may  seem  rash  to  advance  even  this  tentative 
suggestion.  But  yet  there  are  one  or  two  known  facts  which  seem 
to  bear  upon  the  question,  and  to  indicate  at  least  the  manner  in 
which  a  varying  resistance  to  expansion  may  affect  the  velocity 
of  growth.  For  instance,  it  has  been  shewn  by  Frazee*  that 
electrical  stimulation  of  tadpoles,  with  small  current  density  and 
low  voltage,  increases  the  rate  of  regenerative  growth.  As  just 
such  an  electrification  would  tend  to  lower  the  surface-tension,  and 
accordingly  decrease  the  external  resistance,  the  experiment  would 
seem  to  support,  in  some  slight  degree,  the  suggestion  which  I  have 
made. 

To  another  important  aspect  of  regeneration  we  can  do  no  more 
than  allude.  The  Planarian  worms  rival  Hydra  itself  in  their  powers 
of  regeneration;  and  in  both  cases  even  small  bits  of  the  animal 
are  likely  to  include  endoderm  cells  capable  of  intracellular  digestion, 
whereby  the  fragment  is  enabled  to  live  and  to  grow.  Now  if  a 
Planarian  worm  be  cut  in  separate  pieces  and  these  be  suffered  to 
grow  and  regenerate,  they  do  so  in  a  definite  and  orderly  way ;  that 
part  of  a  shce  or  fragment  which  had  been  nearer  tq  the  original 
head  will  develop  a  head,  and  a  tail  will  be  regenerated  at  the 
opposite  end  of  the  same  fragment,  the  end  which  Had  been  tailward 
in  the  beginning;  the  amputated  fragments  possess  sides  and  ends, 
a  front  end  and  a  hind  end,  like  the  entire  worm;  in  short,  they 
retain  their  polarity.  This  remarkable  discovery  is  due  to  Child, 
who  has  amplified  and  extended  it  in  various  instructive  ways. 
The  existence  of  two  poles,  positive  and  negative,  implies  a 
"gradient"  between  them.  It  means  that  one  part  leads  and 
another  follows;  that  one  part  is  dominant,  or  prepotent  over  the 
rest,  whether  in  regenerative  growth  or  embryonic  development. 

We  may  summarise,  as  follows,  the  main  results  of  the  foregoing 
discussion : 

(1)   Except  in  certain  minute  organisms,  whose  form  (hke  that 

*  Journ.  Ezper.  ZooL  vii,  p.  457,  1909. 


Ill]  SUMMARY  AND  CONCLUSION  283 

of  a  drop  of  water)  is  due  to  the  direct  action  of  the  molecular  forces, 
we  may  look  upon  the  form  of  an  organism  as  a  "  function  of  growth," 
or  a  direct  consequence  of  growth  whose  rate  varies  in  its  different 
directions.  In  a  newer  language  we  might  call  the  form  of  an 
organism  an  "event  in  space-time,"  and  not  merely  a  "configuration 
in  space." 

(2)  Growth  varies  in  rate  in  an  orderly  way,  or  is  subject,  like 
other  physiological  activities,  to  definite  "laws."  The  rates  differ 
in  degree,  or  form  "gradients,"  from  one  point  of  an  organism  to 
another;  the  rates  in  different  parts  and  in  different  directions 
tend  to  maintain  more  or  less  constant  ratios  to  one  another  in 
each  organism ; ,  and  to  the  regularity  and  constancy  of  these  relative 
rates  of  growth  is  due  the  fact  that  the  form  of  the  organism  is  in 
general  regular  and  constant. 

(3)  Nevertheless,  the  ratio  of  velocities  in  different  directions  is 
not  absolutely  constant,  but  tends  to  alter  in  course  of  time,  or  to 
fluctuate  in  an  orderly  way;  and  to  these  progressive  changes  are 
due  the  changes  of  form  which  accompany  development,  and  the 
slower  changes  which  continue  perceptibly  in  after  hfe. 

(4)  Rate  of  growth  depends  on  the  age  of  the  organism.  It  has 
a  maximum  somewhat  early  in  hfe,  after  which  epoch  of  maximum 
it  slowly  declines. 

(5)  Rate  of  growth  is  directly  affected  by  temperature,  and  by 
other  physical  conditions:  the  influence  of  temperature  being 
notably  large  in  the  case  of  cold-blooded  or  "  poecilothermic " 
animals.  Growth  tends  in  these  latter  to  be  asymptotic,  becoming 
slower  but  never  ending  with  old  age. 

(6)  It  is  markedly  affected,  in  the  way  of  acceleration  or  retarda- 
tion, at  certain  physiological  epochs  of  hfe,  such  as  birth,  puberty 
or  metamorphosis. 

(7)  Under  certain  circumstances,  growth  may  be  negative,  the 
organism  growing  smaller;  and  such  negative  growth  is  a  common 
accompaniment  of  metamorphosis,  and  a  frequent  concomitant  of 
old  age. 

(8)  The  phenomenon  of  regeneration  is  associated  with  a  large 
transitory  increase  in  the  rate  of  growth  (or  acceleration  of  growth) 
in  the  region  of  injury;  in  other  respects  regenerative  growth  is 
similar  to  ordinary  growth  in  all  its  essential  phenomena. 


284  THE  RATE   OF  GROWTH  [ch. 

In  this  discussion  of  growth,  we  have  left  out  of  account  a  vast 
number  of  processes  or  phenomena  in  the  physiological  mechanism 
of  the  body,  by  which  growth  is  effected  and  controlled.  We  have 
dealt  with  growth  in  its  relation  to  magnitude,  and  to  that  relativity 
of  magnitudes  which  constitutes  form;  and  so  we  have  studied  it 
as  a  phenomenon  which  stands  at  the  beginning  of  a  morphological, 
rather  than  at  the  end  of  a  physiological  enquiry.  Under  these 
restrictions,  we  have  treated  it  as  far  as  possible,  or  in  such  fashion 
as  our  present  knowledge  permits,  on  strictly  physical  lines.  That 
is  to  say,  we  rule  "heredity"  or  any  such  concept  out  of  our  present 
account,  however  true,  however  important,  however  indispen- 
sable in  another  setting  of  the  story,  such  a  concept  may  be. 
In  physics  "on  admet  que  I'etat  actuel  du  monde  ne  depend  que  du 
passe  le.  plus  proche,  sans  etre  influence,  pour  ainsi  dire,  par  le 
souvenir  d'un  passe  lointain*."  This  is  the  concept  to  which  the 
differential  equation  gives  expression;  it  is  the  step  which  Newton 
took  when  he  left  Kepler  behind. 

In  all  its  aspects,  and  not  least  in  its  relation  to  form,  the  growth 
of  organisms  has  many  analogies,  some  close,  some  more  remote, 
among  inanimate  things.  As  the  waves  grow  when  the  winds  strive 
with  the  other  forces  which  govern  the  movements  of  the  surface 
of  the  sea,  as  the  heap  grows  when  we  pour  corn  out  of  a  sack,  as 
the  crystal  grows  when  from  the  surrounding  solution  the  proper 
molecules  fall  into  their  appropriate  places:  so  in  all  these  cases, 
very  much  as  in  the  organism  itself,  is  growth  accompanied  by 
change  of  form,  and  by  a  development  of  definite  shapes  and 
contours.  And  in  these  cases  (as  in  all  other  mechanical  phenomena), 
we  are  led  to  equate  our  various  magnitudes  with  time,  and  so  to 
recognise  that  growth  is  essentially  a  question  of  rate,  or  of  velocity. 

The  diiferences  of  form,  and  changes  of  form,  which  are  brought 
about  by  varying  rates  (or  "laws")  of  growth,  are  essentially  the 
same  phenomenon  whether  they  be  episodes  in  the  life-history  of 
the  individual,  or  manifest  themselves  as  the  distinctive  charac- 
teristics of  what  we  call  separate  species  of  the  race.  From  one 
form,  or  one  ratio  of  magnitude,  to  another  there  is  but  one  straight 
and  direct  road  of  transformation,  be  the  journey  taken  fast  or 

*  Cf.  H.  Poincare,  La  physique  generale  et  la  physique  mathematique,  Rev. 
gin.  des  Sciences,  xi,  p.  1167,  1900. 


Ill]  SUMMARY  AND  CONCLUSION  285 

slow;  and  if  the  transformation  take  place  at  all,  it  will  in  all 
likelihood  proceed  in  the  self-same  way,  whether  it  occur  within 
the  Ufetime  of  an  individual  or  during  the  long  ancestral  history  of 
a  race.  No  small  part  of  what  is  known  as  Wolff's  or  von  Baer*s 
law,  that  the  individual  organism  tends  to  pass  through  the  phases 
characteristic  of  its  ancestors,  or  that  the  life-history  of  the  individual 
tends  to  recapitulate  the  ancestral  history  of  its  race,  lies  wrapped 
up  in  this  simple  account  of  the  relation  between  growth  and  form. 
But  enough  of  this  discussion.  Let  us  leave  for  a  while  the 
subject  of  the  growth  of  the  organism,  and  attempt  to  study  the 
conformation,  within  and  without,  of  the  individual  cell. 


CHAPTER  IV 

ON    THE    INTERNAL    FORM    AND   STRUCTURE 
OF   THE    CELL 

In  the  early  days  of  the  cell-theory,  a  hundred  years  ago,  Goodsir 
was  wont  to  speak  of  cells  as  "centres  of  growth"  or  "centres  of 
nutrition,"  and  to  consider  them  as  essentially  "centres  of  force*". 
He  looked  forward  to  a  time  when  the  forces  connected  with  the 
cell  should  be  particularly  investigated :  when,  that  is  to  say,  minute 
anatomy  should  be  studied  in  its  dynamical  aspect.  "When  this 
branch  of  enquiry,"  he  says,  "shall  have  been  opened  up,  we  shall 
expect  to  have  a  science  of  organic  forces,  having  direct  relation 
to  anatomy,  the  science  of  organic  forms."  And  likewise,  long 
afterwards,  Giard  contemplated  a  science  of  morphodynatnique — but 
still  looked  upon  it  as  forming  so  guarded  and  hidden  a  "territoire 
scientifique,  que  la  plupart  des  naturalistes  de  nos  jours  ne  le  verront 
que  comme  Moise  vit  la  terre  promise,  seulement  de  loin  et  sans 
pouvoir  y  entrerf ." 

To  the  external  forms  of  cells,  and  to  the  forces  which  produce 
and  modify  these  forms,  we  shall  pay  attention  in  a  later  chapter. 
But  there  are  forms  and  configurations  of  matter  within  the  cell 
which  also  deserve  to  be  studied  with  due  regard  to  the  forces, 
known  or  unknown,  of  whose  resultant  they  are  the  visible 
expression. 

*  Anatomical  and  Pathological  Observations,  p.  3,  1845;  Anatomical  Memoirs, 
II,  p.  392,  1868.  This  was  a  notable  improvement  on  the  "kleine  wirkungsfahige 
Zentren  oder  Elementen"  of  the  Cellularpathologie.  Goodsir  seems  to  have  been 
seeking  an  analogy  between  the  living  cell  and  the  physical  atom,  which  Faraday, 
following  Boscovich,  had  been  speaking  of  as  a  centre  of  force  in  the  very  year 
before  Goodsir  published  his  Observations:  see  Faraday's  Speculations  concerning 
Electrical  Conductivity  and  the  Nature  of  Matter,  1844.  For  Newton's  "molecules" 
had  been  turned  by  his  successors  into  material  points;  and  it  was  Boscovich  (in 
1758)  who  first  regarded  these  material  points  as  mere  persistent  centres  of  force. 
It  was  the  same  fertile  conception  of  a  centre  of  force  which  led  Rutherford,  later 
on,  to  the  discovery  of  the  nucleus  of  the  atom. 

t  A.  Giard,  L'oeuf  et  les  debuts  de  revolution,  Bull.  Sci.  du  Nord  de  la  Fr.  vm, 
pp.  252-258,  1876. 


CH.  IV]  THE  CELL  THEORY      .  287 

In  the  long  interval  since  Goodsir's  day,  the  visible  structure, 
the  conformation  and  configuration,  of  the  cell,  has  been  studied 
far  more  abundantly  than  the  purely  dynamic  problems  which  are 
associated  therewith.  The  overwhelming  progress  of  microscopic 
observation  has  multipHed  our  knowledge  of  cellular  and  intra- 
cellular structure ;  and  to  the  multitude  of  visible  structures  it  has 
been  often  easier  to  attribute  virtues  than  to  ascribe  intelUgible 
functions  or  modes  of  action.  But  here  and  there  nevertheless, 
throughout  the  whole  hteratiire  of  the  subject,  we  find  recognition 
of  the  inevitable  fact  that  dynamical  problems  lie  behind  the 
morphological  problems  of  the  cell. 

Biitschli  pointed  out  sixty  years  ago,  with  emphatic  clearness, 
the  failure  of  morphological  methods  and  the  need  for  physical 
methods  if  we  were  to  penetrate  deeper  into  the  essential  nature  of 
the  cell*.  And  such  men  as  Loeb  and  Whitman,  Driesch  and  Roux, 
and  not  a  few  besides,  have  pursued  the  same  train  of  thought  and 
similar  methods  of  enquiry. 

Whitman t,  for  instance,  puts  the  case  in  a  nutshell  when,  in 
speaking  of  the  so-called  "  caryokinetic "  phenomena  of  nuclear 
division,  he  reminds  us  that  the  leading  idea  in  the  term  ''caryo- 
kinesis''  is  ynotion — "motion  viewed  as  an  exponent  of  forces 
residing  in,  or  acting  upon,  the  nucleus.  It  regards  the  nucleus 
as  a  seat  of  energy,  which  displays  itself  in  phenomena  of  motion X-^' 

In  short  it  would  seem  evident  that,  except  in  relation  to  a 
dynamical  investigation,  the  mere  study  of  cell  structure  has  but 

*  Entwickelungsvorgdnge  der  Eizelle,  1876;  Investigations  on  Microscopic  Foams 
and  Protoplasm,  p.  1,  1894. 

t  Journ.  Morphology,  i,  p.  229,  1887. 

t  While  it  has  been  very  common  to  look  upon  the  phenomena  of  mitosis  as 
sufficiently  explained  by  the  results  towards  which  they  seem  to  lead,  we  may  find 
here  and  there  a  strong  protest  against  this  mode  of  interpretation.  The  following 
is  a  case  in  point:  ''On  a  tente  d'etablir  dans  la  mitose  dite  primitive  plusieurs 
categories,  plusieurs  types  de  mitose.  On  a  choisi  le  plus  souvent  comme  base 
de  ces  systemes  des  concepts  abstraits  et  teleologiques :  repartition  plus  ou  moins 
exacte  de  la  chromatine  entre  les  deux  noyaux-fils  suivant  qu'il  y  a  ou  non  des 
chromosomes  (Dangeard),  distribution  particuliere  et  signification  dualiste  des 
substances  nucleaires  (substance  kinetique  et  substance  generative  ou  hereditaire, 
Ilartmann  et  ses  eleves),  etc.  Pour  moi  tous  ces  essais  sont  a  rejeter  categorique- 
ment  a  cause  de  leur  caractere  finaliste;  de  plus,  ils  sont  construits  sur  des  concepts 
non  demontres,  et  qui  parfois  representent  des  generalisations  absolument  erronees.'' 
A.  Alexeieflf,  Archiv  fiir  Protistenkunde,  xix,  p.  344,  1913. 


288  ON  THE  INTERNAL  FORM  [ch. 

little  value  of  its  own.  That  a  given  cell,  an  ovum  for  instance, 
contains  this  or  that  visible  substance  or  structure,  germinal  vesicle 
or  germinal  spot,  chromatin  or  achromatin,  chromosomes  or  centro- 
somes,  obviously  gives  no  explanation  of  the  activities  of  the  cell. 
And  in  all  such  hypotheses  as  that  of  "pangenesis,"  in  all  the 
theories  which  attribute  specific  properties  to  micellae,  chromosomes, 
idioplasts,  ids,  or  other  constituent  particles  of  protoplasm  or  of 
the  cell,  we  are  apt  to  fall  into  the  error  of  attributing  to  matter 
what  is  due  to  energy  and  is  manifested  in  force:  or,  more  strictly 
speaking,  of  attributing  to  material  particles  individually  what  is 
due  to  the  energy  of  their  collocation. 

The  tendency  is  a  very  natural  one,  as  knowledge  of  structure 
increases,  to  ascribe  particular  virtues  to  the  material  structures 
themselves,  and  the  error  is  one  into  which  the  disciple  is  Ukely 
to  fall  but  "of  which  we  need  not  suspect  the  master-mind.  The 
dynamical  aspect  of  the  case  was  in  all  probability  kept  well  in  view 
by  those  who,  Hke  Goodsir  himself,  first  attacked  the  problem  of 
the  cell  and  originated  our  conceptions  of  its  nature  and  functions*. 

If  we  speak,  as  Weismann  and  others  speak,  of  an  "hereditary 
sitbstance,''  a  substance  which  is  spHt  off  from  the  parent-body,  and 
which  hands  on  to  the  new  generation  the  characteristics  of  the  old, 
we  can  only  justify  our  mode  of  speech  by  the  assumption  that  that 
particular  portion  of  matter  is  the  essential  vehicle  of  a  particular 
charge  or  distribution  of  energy,  in  which  is  involved  the  capabihty 
of  producing  motion,  or  of  doing  "work."  For,  as  Newton  said, 
to  tell  us  that  a  thing  "is  endowed  with  an  occult  specific  quahtyf, 
by  which  it  acts  and  produces  manifest  effects,  is  to  tell  us  nothing ; 
but  to  derive  two  or  three  general  principles  of  motion  {   from 

*  See  also  {int.  al.)  R.  S.  Lillie's  papers  on  the  physiology  of  cell-division  in  the 
Journ.  Exper.  Physiology;  especially  No.  vi,  Rhythmical  changes  in  the  resistance 
of  the  dividing  sea-urchin  egg,  ibid,  xvi,  pp.  369-402,  1916. 

f  Such  as  the  vertu  donnitive  which  accounts  for  the  soporific  action  of  opium. 
We  are  now  more  apt.  as  Le  Dantec  says,  to  substitute  for  this  occult  quality  the 
hypothetical  substance  dormitin. 

X  This  is  the  old  philosophic  axiom  writ  large:  Ignorato  motu,  ignoratur  -natura; 
which  again  is  but  an  adaptation  of  Aristotle's  phrase,  17  dpx^  ^V^  Kivrjcrecxjs,  as 
equivalent  to  the  "Efiicient  Cause."  FitzGerald  holds  that  "all  explanation 
consists  in  a  description  of  underlying  motions"  {Scientific  Writings,  1902,  p.  385); 
and  Oliver  Lodge  remarked,  "You  can  move  Matter;  it  is  the  only  thing  you  can 
do  to  it." 


IV]  AND  STRUCTURE  OF  THE  CELL  289 

phenomena  would  be  a  very  great  step  in  philosophy,  though  the 
causes  of  those  principles  were  not  yet  discovered."  The  things 
which  we  see  in  the  cell  are  less  important  than  the  actions  which 
we  recognise  in  the  cell;  and  these  latter  we  must  especially, 
scrutinise,  in  the  hope  of  discovering  how  far  they  may  be  attributed 
to  the  simple  and  well-known  physical  forces,  and  how  far  they  be 
relevant  or  irrelevant  to  the  phenomena  which  we  associate  with, 
and  deem  essential  to,  the  manifestation  of  life.  It  may  be  that  in 
this  way  we  shall  in  time  draw  nigh  to  the  recognition  of  a  specific 
and  ultimate  residuum. 

And  lacking,  as  we  still  do  lack,  direct  knowledge  of  the 
actual  forces  inherent  in  the  cell,  we  may  yet  learn  something 
of  their  distribution,  if  not  also  of  their  nature,  from  the 
outward  and  inward  configuration  of  the  cell  and  from  the 
changes  taking  place  in  this  configuration;  that  is  to  say  from 
the  movements  of  matter,  the  kinetic  phenomena,  which  the  forces 
in  action  set  up. 

The  fact  that  the  germ-cell  develops  into  a  very  complex  structure 
is  no  absolute  proof  that  the  cell  itself  is  structurally  a  very  com- 
phcated  mechanism :  nor  yet  does  it  prove,  though  this  is  somewhat 
less  obvious,  that  the  forces  at  work  or  latent  within  it  are  especially 
numerous  and  complex.  If  we  blow  into  a  bowl  of  soapsuds  and 
raise  a  great  mass  of  many-hued  and  variously  shaped  bubbles,  if 
we  explode  a  rocket  and  watch  the  regular  and  beautiful  configura- 
tion of  its  falhng  streamers,  if  we  consider  the  wonders  of  a  Hmestone 
cavern  which  a  filtering  stream  has  filled  with  stalactites,  we  soon 
perceive  that  in  all  these  cases  we  have  begun  with  an  initial  system 
of  very  slight  complexity,  whose  structure  in  no  way  foreshadowed 
the  result,  and  whose  comparatively  simple  intrinsic  forces  only 
play  their  part  by  complex  interaction  with  the  equally  simple 
forces  of  the  surrounding  medium.  In  an  earlier  age,  men  sought 
for  the  visible  embryo,  even  for  the  homunculus,  within  the  repro- 
ductive cells;  and  to  this  day  we  scrutinise  these  cells  for  visible 
structure,  unable  to  free  ourselves  from  that  old  doctrine  of 
' '  pre-f ormation  * . " 

Moreover,  the  microscope  seemed  to  substantiate  the  idea  (which 

*  As  when  Nageli  concluded  that  the  organism  is,  in  a  certain  sense,  "vorge- 
bildet";   Beitr.  zur  wiss.  Botanik,  ii,  1860. 


290  ON  THE  INTERNAL  FORM  [ch. 

we  may  trace  back  to  Leibniz*  and  to  Hobbest),  that  there  is  no 
Hmit  to  the  mechanical  complexity  which  we  may  postulate  in  an 
organism,  and  no  limit,  therefore,  to  the  hypotheses  which  we  may 
rest  thereon.  But  no  microscopical  examination  of  a  stick  of  sealing- 
wax,  no  study  of  the  material  of  which  it  is  composed,  can  enlighten 
us  as  to  its  electrical  manifestations  or  properties.  Matter  of  itself 
has  no  power  to  do,  to  make,  or  to  become:  it  is  in  energy  that 
all  these  potentiaUties  reside,  energy  invisibly  associated  with  the 
material  system,  and  in  interaction  with  the  energies  of  the 
surrounding  universe. 

That  ''function  presupposes  structure"  has  been  declared  an 
accepted  axiom  of  biology.  Who  it  was  that  so  formula  ted  the 
aphorism  I  do  not  know;  but  as  regards  the  structure  of  the  cell 
it  harks  back  to  Briicke,  with  whose  demand  for  a  mechanism,  or 
an  organisation,  within  the  cell  histologists  have  ever  since  been 
trying  to  comply  J.  But  unless  we  mean  to  include  thereby 
invisible,  and  merely  chemicu,l  or  molecular,  structure,  we  come  at 
once  on  dangerous  ground.  For  we  have  seen  in  a  former  chapter 
that  organisms  are  known  of  magnitudes  so  nearly  approaching  the 
molecalar,  that  everything  which  the  morphologist  is  accustomed  to 
conceive  as  "structure"  has  become  physically  impossible;  and 
recent  research  tends  to  reduce,  rather  than  to  extend,  our  con- 
ceptions of  the  visible  structure  necessarily  inherent  in  living 
protoplasm §.     The  microscopic  structure  which  in  the  last  resort 

*  "La  matiere  arrangee  par  une  sagesse  divine  doit  etre  essentiellement  organisee 
partout. .  .il  y  a  machine  dans  les  parties  de  la  machine  naturelle  a  I'infini."  Siir  le 
principe  de  la  Vie,  p.  431  (Erdmann).  This  is  the  very  converse  of  the  doctrine 
of  the  Atomists,  who  could  not  conceive  a  condition  "w6/  dimidiae  partis  pars 
semper  hahehit  Dimidiam  partem,  nee  res  praefiniet  ulla.'' 

t  Cf.  an  interesting  passage  from  the  Elements  (i,  p.  445,  Molesworth's  edit.), 
quoted  by  Owen,  Hunterian  Lectures  on  the  Invertebrates,  2nd  ed.  pp.  40,  41,  1855. 

%  "Wir  miissen  deshalb  den  lebenden  Zellen,  abgesehen  von  der  Molekular- 
structur  der  organischen  Verbindungen  welche  sie  enthalt,  noch  eine  andere  und 
in  anderer  Weise  complicirte  Structur  zuschreiben,  und  diese  es  ist  welche  wir 
mit  dem  Namen  Organisation  bezeichnen,"  Briicke,  Die  Elementarorganismen, 
Wiener  Sitzungsber.  xliv,  1861,  p.  386;  quoted  by  Wilson,  The  Cell,  etc.,  p.  289. 
Cf.  also  Hardy,  Journ.  Physiol,  xxiv,  1899,  p.  159. 

§  The  term  protoplasm  was  first  used  by  Purkinje,  about  1839  or  1840  (cf. 
Reichert,  Arch.  f.  Anat.  u.  Physiol.  1841).  But  it  was  better  defined  and  more 
strictly  used  by  Hugo  von  Mohl  in  his  paper  Ueber  die  Saftbewegung  im  Inneren 
der  Zellen,  Botan.  Zeitung,  iv,  col.  73-78,  89-94,  1846. 


IV]  AND  STRUCTURE  OF  THE  CELL  291 

or  in  the  simplest  cases  it  seems  to  sheWj  is  that  of  a  more  or  less 
viscous  colloid,  or  rather  mixtm:e  of  colloids,  and  nothing  more. 
Now,  as  Clerk  Maxwell  puts  it  in  discussing  this  very  problem, 
"one  material  system  can  differ  from  another  only  in  the  configura- 
tion and  motion  which  it  has  at  a  given  instant*."  If  we  cannot 
assume  differences  in  structure  or  configuration,  we  must  assume 
differences  in  motion,  that  is  to  say  in  energy.  And  if  we  cannot 
do  this,  then  indeed  we  are  thrown  back  upon  modes  of  reasoning 
unauthorised  in  physical  science,  and  shall  find  ourselves  constrained 
to  assume,  or  to  "admit,  that  the  properties  of  a  germ  are  not  those 
of  a  purely  material  system." 

But  we  are  by  no  means  necessarily  in  this  dilemma.  For  though 
we  come  perilously  near  to  it  when  we  contemplate  the  lowest 
orders  of  magnitude  to  which  hfe  has  been  attributed,  yet  in  the 
case  of  the  ordinary  cell,  or  ordinary  egg  or  germ  which  is  going 
to  develop  into  a  complex  organism,  if  we  have  no  reason  to  assume 
or  to  beheve  that  it  comprises  an  intricate  "mechanism,"  we  mgiy 
be  quite  sure,  both  on  direct  and  indirect  evidence,  that,  hke  the 
powder  in  our  rocket,  it  is  very  heterogeneous  in  its  structure. 
It  is  a  mixture  of  substances  of  various  kinds,  more  or  less  fluid, 
more  or  less  mobile,  influenced  in  various  ways  by  chemical,  electrical, 
osmotic  and  other  forces,  and  in  their  admixture  separated  by  a 
multitude  of  surfaces  or  boundaries,  at  which  these  or  certain  of 
these  forces  are  made  manifest. 

Indeed,  such  an  arrangement  as  this  is  already  enough  to  con- 
stitute a  "mechanism";  for  we  must  be  very  careful  not  to  let  our 
physical  or  physiological  concept  of  mechanism  be  narrowed  to  an 
interpretation  of  the  term  derived  from  the  comphcated  contrivances 
of  himaan  skill.  From  the  physical  point  of  view,  we  understand 
by  a  "mechanism"  w^hatsoever  checks  or  controls,  and  guides  into 
determinate  paths,  the  workings  of  energy:  in  other  words,  what- 
soever leads  in  the  degradation  of  energy  to  its  manifestation  in 
some  form  of  work,  at  a  stage  short  of  that  ultimate  degradation 
which  lapses  in  uniformly  diffused  heat.  This,  as  Warburg  has  well 
explained,  is  the  general  effect  or  function  of  the  physiological 
machine,  and  in  particular  of  that  part  of  it  which  we  call  "cell- 

*  Precisely  as  in  the  Lueretian  concursus,  motus,  ordo,  positura,  figurae,  whereby 
bodies  mutato  ordine  mutant  naturam. 


292  ON  THE  INTERNAL  FORM  [ch. 

structure*."  The  normal  muscle-cell  is  something  which  turns 
energy,  derived  from  oxidation,  into  work;  it  is  a  mechanism  which 
arrests  and  utihses  the  chemical  energy  of  oxidation  in  its  downward 
course;  but  the  same  cell  when  injured  or  disintegrated  loses  its 
"usefulness,"  and  sets  free  a  greatly  increased  proportion  of  its 
energy  in  the  form  of  heat.  It  was  a  saying  of  Faraday's,  that 
"even  a  Hfe  is  but  a  chemical  act  prolonged.  If  death  occur,  the 
more  rapidly  oxygen  and  the  affinities  run  on  to  their  final  state  f." 
Very  great  and  wonderful  things  are  done  by  means  of  a 
mechanism  (whether  natural  or  artificial)  of  extreme  simphcity. 
A  pool  of  water,  by  virtue  of  its  surface,  is  an  admirable  mechanism 
for  the  making  of  waves ;  with  a  lump  of  ice  in  it,  it  becomes  an 
efficient  and  self-contained  mechanism  for  the  making  of  currents. 
Music  itself  is  made  of  simple  things — a  reed,  a  pipe,  a  string. 
The  great  cosmic  mechanisms  are  stupendous  in  their  simphcity; 
and,  in  point  of  fact,  every  great  or  little  aggregate  of  heterogeneous 
matter  (not  identical  in  "phase")  involves,  ipso  facto,  the  essentials 
of  a  mechanism.  Even  a  non-living  colloid,  from  its  intrinsic  hetero- 
geneity, is  in  this  sense  a  mechanism,  and  one  in  which  energy  is 
manifested  in  the  movement  and  ceaseless  rearrangement  of  the 
constituent  particles.  For  this  reason  Graham  speaks  somewhere 
or  other  of  the  colloid  state  as  "the  dynamic  state  of  matter";  in 
the  same  philosopher's' phrase,  it  possesses  ^' energia%.'^ 

Let  us  turn  then  to  consider,  briefly  and  diagrammatically,  the 
structure  of  the  cell,  a  fertilised  germ-cell  or  ovum  for  instance,  not 
in  any  vain  attempt  to  correlate  this  structure  with  the  structure 
or  properties  of  the  resulting  and  yet  distant  organism ;  but  merely 
to  see  how  far,  by  the  study  of  its  form  and  its  changing  internal 
configuration,  we  may  throw  hght  on  certain  forces  which  are  for 
the  time  being  at  work  within  it. 

We  may  say  at  once  that  we  can  scarcely  hope  to  learn  more  of 
these  forces,  in  the  first  instance,  than  a  few  facts  regarding  their 

*  Otto  Warburg,  Beitrage  zur  Physiologie  der  Zelle,  insbesondere  iiber  die 
Oxidationsgeschwindigkeit  in  Zellen;  in  Asher-Spiro's  Ergebnisse  der  Physiologie, 
XIV,  pp.  253-337,  1914  (see  p.  315). 

t  See  his  Life  by  Bence  Jones,  ii,  p.  299. 

X  Both  phrases  occur,  side  by  side,  in  Graham's  classical  paper  on  Liquid 
diffusion  applied  to  analysis,  Phil.  Trans,  cli,  p.  184,  1861;  Chem.  and  Phys. 
Researches  (ed.  Angus  Smith),  1876,  p.  554. 


iv]  AND  STRUCTURE  OF  THE  CELL  293 

direction  and  magnitude;  tli£  nature  and  specific  identity  of  the 
force  or  forces  is  a  very  different  matter.  This  latter  problem  is 
likely  to  be  difficult  of  elucidation,  for  the  reason,  among  others, 
that  very  different  forces  are  often  much  alike  in  their  outward  and 
visible  manifestations.  So  it  has  come  to  pass  that  we  have  a 
multitude  of  discordant  hypotheses  as  to  the  nature  of  the  forces 
acting  within  the  cell,  and  producing  in  cell  division  the  "caryo- 
kinetic"  figures  of  which  we  are  about  to  speak.  One  student  may, 
like  Rhumbler,  choose  to  account  for  them  by  an  hj^othesis  of 
mechanical  traction,  acting  on  a  reticular  web  of  protoplasm*; 
another,  hke  Leduc,  may  shew  us  how  in  many  of  their  most  striking 
features  they  may  be  admirably  simulated  by  salts  diffusing  in  a 
colloid  medium;  others,  hke  Lamb  and  Graham  Cannon,  have 
compared  them  to  the  stream-hnes  produced  and  the  field  of  force 
set  up  by  bodies  vibrating  in  a  fluid;  others,  like  Gallardof  and 
Rhumbler  in  his  earher  papers  J,  insisted  on  their  resemblance  to 
certain  phenomena  of  electricity  and  magnetism  §;  while  Hartog 
believed  that  the  force  in  question  is  only  analogous  to  these,  and 
has  a  specific  identity  of  its  own||.  All  these  conflicting  views  are 
of  secondary  importance,  so  long  as  we  seek  only  to  account  for 
certain  configurations  which  reveal  the  direction,  rather  than  the 
nature,  of  a  force.  One  and  the  same  system  of  lines  of  force  may 
appear  in  a  field  of  magnetic  or  of  electrical  energy,  of  the  osmotic 
energy  of  diffusion,  of  the  gravitational  energy  of  a  flowing  stream. 
In  short,  we  may  expect  to  learn  something  of  the  pure  or  abstract 
dynamics  long  before  we  can  deal  with  the  special  physics  of  the 

*  L.  Rhumbler,  Mechanische  Erklarung  der  Aehnlichkeit  zwischen  magne- 
tischen  Kraftliniensystemen  und  Zelltheilungsfiguren,  Arch.  f.  Entw.  Mech.  xv, 
p.  482,  1903. 

t  A.  Gallardo,  Essai  d'interpretation  des  figures  caryocinetiques,  Armies  del 
Museo  de  Buenos- Aires  (2),  ii,  1896;   Arch.  f.  Entw.  Mech.  xxvm,  1909,  etc. 

X  Arch.f.  Entw.  Mech.  ill,  iv,  1896-97. 

§  On  various  theories  of  the  mechanism  of  mitosis,  see  (e.g.)  Wilson,  The  Cell 
in  Development,  etc.;  Meves,  Zelltheilung,  in  Merkel  u.  Bonnet's  Ergehnisse  der 
Anatomic,  etc.,  vii,  viii,  1897-98;  Ida  H.  Hyde,  Amer.  Journ.  Physiol,  xii,  pp.  241- 
275,  1905;  and  especially  A.  Prenant,  Theories  et  interpretations  physiques  de 
la  mitose,  Journ.  de  VAnat.  et  Physiol,  xlvi,  pp.  511-578,  1910.  See  also  A.  Conard, 
Sur  le  mecanisme  de  la  division  cellulaire,  et  sur  les  bases,  morphologiques  de  la 
Cytologie,  Bruxelles,  1939:  a  work  which  I  find  hard  to  follow. 

II  M.  Hartog,  Une  force  nouvelle:  le  mitokinetisme,  C.R.  11  Juli  1910;  Arch.  f. 
Entw.  Mech.  xxvii,  pp.  141-145,  1909;    cf.  ibid,  xl,  pp.  33-64,  1914. 


294  ON  THE  INTERNAL  FORM  [ch. 

cell.  For  indeed,  just  as  uniform  expansion  about  a  single  centre, 
to  whatsoever  physical  cause  it  may  be  due,  will  lead  to  the  con- 
figuration of  a  sphere,  so  will  any  two  centres  or  foci  of  potential 
(of  whatsoever  kind)  lead  to  the  configurations  with  which  Faraday 
first  made  us  famihar  under  the  name  of  "lines  of  force*";  a^d 
this  is  as  much  as  to  say  that  the  phenomenon,  though  physical  in  the 
concrete,  is  in  the  abstract  purely  mathematical,  and  in  its  very  essence 
is  neither  more  nor  less  than  a  property  of  three-dimensional  space. 

But  as  a  matter  of  fact,  in  this  instance,  that  is  to  say  in  trying 
to  explain  the  leading  phenomena  of  the  caryokinetic  division  of 
the  cell,  we  shall  soon  perceive  that  any  explanation  which  is  based, 
like  Rhumbler's,  on  mere  mechanical  traction,  is  obviously  inade- 
quate, and  we  shall  find  ourselves  limited  to  the  hypothesis  of  some 
polarised  and  polarising  force,  such  as  we  deal  with,  for  instance, 
in  magnetism  or  electricity,  or  in  certain  less  familiar  phenomena 
of  hydrodynamics.  Let  us  speak  first  of  the  cell  itself,  as  it  appears 
in  a  state  of  rest,  and  let  us  proceed  afterwards  to  study  the  more 
active  phenomena  which  accompany  its  division. 

Our  typical  cell  is  a  spherical  body;  that  is  to  say,  the  uniform 
surface-tension  at  its  boundary  is  balanced  by  the  outward  resistance 
of  uniform  forces  within.  But  at  times  the  surface-tension  may  be 
a  fluctuating  quantity,  as  when  it  produces  the  rhythmical  con- 
tractions or  "Ransom's  waves "f  on  the  surface  of  a  trout's  egg;  or 
again,  the  surface-tension  may  be  locally  unequal  and  variable,  giving 
rise  to  an  amoeboid  figure,  as  in  the  egg  of  HydraX- 

Within  the  cell  is  a  nucleus  or  germinal  vesicle,  also  spherical, 

*  The  configurations,  as  obtained  by  the  usual  experimental  methods,  were 
of  course  known  long  before  Faraday's  day,  and  constituted  the  "convergent  and 
divergent  magnetic  curves"  of  eighteenth  century  mathematicians.  As  Leslie 
said,  in  1821,  they  were  "regarded  with  wonder  by  a  certain  class  of  dreaming 
philosophers,  whp  did  not  hesitate  to  consider  them  as  the  actual  traces  of  an 
invisible  fluid,  perpetually  circulating  between  the  poles  of  the  magnet."  Faraday's 
great  advance  was  to  interpret  them  as  indications  of  stress  in  a  medium — of. 
tension  or  attraction  along  the  lines,  and  of  repulsion  transverse  to  the  lines,  of  the 
diagram. 

t  W.  H.  Ransom,  On  the  ovum  of  osseous  fishes,  Phil.  Trans,  clvii,  pp.  431-502, 
1867  (vide  p.  463  et.  seq.)  (Ransom,  afterwards  a  Nottingham  physician,  was 
Huxley's  friend  and  class-fellow  at  University  College,  and  beat  him  for  the  medal 
in  Grant's  class  of  zoology.) 

X  Cf,  also  the  curiou.s  phenomenon  in  a  dividing  egg  described  as  "spinning" 
by  Mrs  G.  F.  Andrews,  Journ.  Morph.  xii,  pp.  367-389,  1897. 


IV]  AND  STRUCTURE  OF  THE  CELL  295 

and  consisting  of  portions  of  "chromatin,"  aggregated  together 
within  a  more  fluid  drop.  The  fact  has  often  been  commented 
upon  that,  in  cells  generally,  there  is  no  correlation  of  form 
(though  there  apparently  is  of  size)  between  the  nucleus  and  the 
"cytoplasm,"  or  main  body  of  the  cell.  So  Whitman*  remarks 
that  "except  during  the  process  of  division  the  nucleus  seldom 
departs  from  it^  cypical  spherical  form.  It  divides  and  sub-divides, 
ever  returning  to  the  same  round  or  oval  form. .  . .  How  different 
with  the  cell.  It  preserves  the  spherical  form  as  rarely  as  the 
nucleus  departs  from  it.  Variation  in  form  marks  the  beginning 
and  the  end  of  every  important  chapter  in  its  history."  On  simple 
dynamical  grounds,  the  contrast  is  easily  explained.  So  long  as 
the  fluid  substance  of  the  nucleus  is  qualitatively  different  from, 
and  incapable  of  mixing  with,  the  fluid  or  semi-fluid  protoplasm 
surrounding  it,  we  shall  expect  it  to  be,  as  it  almost  always  is,  of 
spherical  form.  For  on  the  one  hand,  it  has  a  surface  of  its  own 
whose  surface-tension  is  presumably  uniform,  and  on  the  other,  it 
is  immersed  in  a  medium  which  transmits  on  all  sides  a  uniform 
fluid  or  "hydrostatic"  pressure f;  thus  the  case  of  the  spherical 
nucleus  is  closely  akin  to  that  of  the  spherical  yolk  within  the 
bird's  egg.  Again,  for  a  similar  reason,  the  contractile  vacuole  of 
a  protozoon  is  spherical {.     It  is  just  a  drop  of  fluid,  bounded  by  a 

*  Whitman,  Journ.  Morph.  ii,  p.  40,  1889. 

t  "Souvent  il  n'y  a  qu'une  separation  physique  entre  le  cytoplasme  et  le  sue 
hucleaire,  comme  entre  deux  liquides  immiscibles,  etc.";  Alexeieff,  8ur  la  mitose 
dite  primitive,  Arch.  f.  Protistenk.  xxix,  p.  357,  1913. 

X  The  appearance  of  "  vacuolation "  is  a  result  of  endosmosis,  or  the  diffusion 
of  a  less  dense  fluid  into  the  denser  plasma  of  the  cell.  But  while  water  is  probably 
taken  up  at  the  surface  of  the  cell  by  purely  passive  osmotic  intake,  a  definite 
"vacuole"  appears  at  a  place  where  osmotic  work  is  being  actively  done.  A  higher 
osmotic  pressure  than  that  of  the  external  medium  is  maintained  within  the  cell, 
but  as  a  "steady  state"  rather  than  a  condition  of  equilibrium,  in  other  words  by 
the  continual  expenditure  of  energy;  and  the  difference  of  pressure  is  at  best  small. 
The  "contractile  vacuole"  bursts  when  it  touches  the  surface  of  the  cell,  and 
bursting  may  be  delayed  by  manipulating  the  vacuole  towards  the  interior.  It 
may  sometimes  burst  towards  the  interior  of  the  cell  through  inequalities  in  its 
own  surface-tension,  and  the  collapsing  vacuole  is  then  apt  to  shew  a  star-shaped 
figure.  The  cause  of  the  higher  osmotic  pressure  within  the  cell  is  a  matter  for 
the  colloid  chemist,  and  cannot  be  discussed  here.  On  the  physiology  of  the 
contractile  vacuole,  see  {int.  al.)  H.  Z.  Gow,  Arch.  f.  Protistenk.  lxxxvii,  pp.  185- 
212,  1936;  J.  Spek,  Einfluss  der  Salze  auf  die  Plasmkolloide  von  Actinosphaerium, 
Acta  Zool.  1921;   J.  A,  Kitching,  Journ.  Exp.  Biology,  xi,  xiii,  xv,  1934-38. 


296  ON  THE  INTERNAL  FORM  [ch. 

uniform  surface-tension,  and  through  whose  boundary-film  diffusion 
is  taking  place;  but  here,  owing  to  the  small  difference  between  the 
fluid  constituting  and  that  surrounding  the  drop,  the  surface-tension 
equihbrium  is  somewhat  unstable;  it  is  apt  to  vanish,  and  the 
rounded  outhne  of  the  drop  disappears,  Uke  a  burst  bubble,  in  a 
moment. 

If,  on  the  other  hand,  the  substance  of  the  cell  acquire  a  greater 
soUdity,  as  for  instance  in  a  muscle-cell,  or  by  reason  of  mucous 
accumulations  in  an  epithehum  cell,  then  the  laws  of  fluid  pressure 
no  longer  apply,  the  pressure  on  the  nucleus  tends  to  become 
unsymmetrical,  and  its  shape  is  modified  accordingly.  Amoeboid 
movements  may  be  set  up  in  the  nucleus  by  anything  which  disturbs 
the  symmetry  of  its  own  surface-tension;  and  where  "nuclear 
material"  is  scattered  in  small  portions  throughout  the  cell  as  in 
many  Rhizopods,  instead  of  being  aggregated  in  a  single  nucleus, 
the  simple  explanation  probably  is  that  the  "phase  difference"  (as 
the  chemists  say)  between  the  nuclear  and  the  protoplasmic  substance 
is  comparatively  shght,  and  the  surface-tension  which  tends  to  keep 
them  separate  is  correspondingly  small*. 

Apart  from  that  invisible  or  ultra-microscopic  heterogeneity 
which  is  inseparable  from  our  notion  of  a  "colloid,"  there  is  a 
visible  heterogeneity  of  structure  within  both  the  nucleus  and  the 
outer  protoplasm.  The  former  contains,  for  instance,  a  rounded 
nucleolus  or  "germinal  spot,"  certain  conspicuous  granules  or 
strands  of  the  peculiar  substance  called  chromatin*)*,  and  a  coarse 
mesh  work  of  a  protoplasmic  material  known  as  "linin"  or  achro- 
matin;  the  outer  protoplasm,  or  cytoplasm,  is  generally  believed 
to  consist  throughout  of  a  sponge-work,  or  rather  alveolar  mesh- 
work,  of  more  and  less  fluid  substances;  it  may  contain  "mito- 
chondria," appearing  in  tissue-cultures  as  small  amoeboid  bodies; 
and  lastly,  there  are  generally  to  be  detected  (in  the  animal,  rarely 
in  the  vegetable  kingdom)  one  or  more  very  minute  bodies,  usually 
in  the  cytoplasm  sometimes  within  the  nucleus,  known  as  the 
centrosome  or  centrosomes. 

*  The  elongated  or  curved  "macronucleus"  of  an  Infusorian  is  to  be  looked 
upon  as  a  single  mass  of  chromatin,  rather  than  as  an  aggregation  of  particles  in 
a  fluid  drop,  as  in  the  case  described.  It  has  a  shape  of  its  own,  in  which  ordinary 
surface-tension  plays  a  very  subordinate  part. 

•j-  First  so-called  by  W.  Flemming,  in  his  Zellsubstanz,  Kern  und  Zelltheilung,  1882. 


IV]  AND  STRUCTURE  OF  THE  CELL  297 

The  morphologist  is  accustomed  to  speak  of  a  "polarity"  of  the 
cell,  meaning  thereby  a  symmetry  of  visible  structure  about  a 
particular  axis.  For  instance,  whenever  we  can  recognise  in  a  cell 
both  a  nucleus  and  a  centrosome,  we  may  consider  a  hne  drawn 
through  the  two  as  the  morphological  axis  of  polarity ;  an  epitheUum 
cell  is  morphologically  symmetrical  about  a  median  axis  passing 
from  its  free  surface  to  its  attached  base.  Again,  by  an  extension 
of  the  term  polarity,  as  is  customary  in  dynamics,  we  may  have 
a  "radial"  polarity,  between  centre  and  periphery;  and  lastly,  we 
may  have  several  apparently  independent  centres  of  polarity  within 
the  single  cell.  Only  in  cells  of  quite  irregular  or  amoeboid  form 
do  we  fail  to  recognise  a  definite  and  symmetrical  polarity.  The 
morphological  polarity  is  accompanied  by,  and  is  but  the  outward 
expression  (or  part  of  it)  of  a  true  dynamical  polarity,  or  distribution 
of  forces;  and  the  hues  of  force  are,  or  may  be,  rendered  visible 
by  concatenation  of  particles  of  matter,  such  as  come  under  the 
influence  of  the  forces  in  action. 

When  hnes  of  force  stream  inwards  from  the  periphery  towards 
a  point  in  the  interior  of  the  cell,  particles  susceptible  of  attraction 
either  crowd  towards  the  surface  of  the  cell  or,  when  retarded  by 
friction,  are  seen  forming  lines  or  "fibrillae"  which  radiate  outwards 
from  the  centre.  In  the  cells  of  columnar  or  cihated  epithehum, 
where  the  sides  of  the  cell  are  symmetrically  disposed  to  their 
neighbours  but  the  free  and  attached  surfaces  are  very  diverse  from 
one  another  in  their  external  relations,  it  is  these  latter  surfaces 
which  constitute  the  opposite  poles;  and  in  accordance  with  the 
parallel  lines  of  force  so  set  up,  we  very  frequently  see  parallel  lines 
of  granules  which  have  ranged  themselves  perpendicularly  to  the 
free  surface  of  the  cell  (cf.  Fig.  149). 

A  simple  manifestation  of  polarity  may  be  well  illustrated  by 
the  phenomenon  of  diffusion,  where  we  may  conceive,  and  may 
automatically  reproduce,  a  field  of  force,  with  its  poles  and  its 
visible  lines  of  equipotential,  very  much  as  in  Faraday's  conception 
of  the  field  of  force  of  a  magnetic  system.  Thus,  in  one  of  Leduc's 
experiments*,  if  we  spread  a  layer  of  salt  solution  over  a  level 
plate  of  glass,  and  let  fall  into  the  middle  of  it  a  drop  of  indian 
ink,  or  of  blood,  we  shall  find  the  coloured  particles  travelling 
*  Thtorie  physico-chimique  de  la  Vie,  1910,  p.  73. 


298  ON  THE  INTERNAL  FORM  [ch 

outwards  from  the  central  "pole  of  concentration"  along  the  lines 
of  diffusive  force,  and  so  mapping  out  for  us  a  "monopolar  field" 
of  diffusion :  and  if  we  set  two  such  drops  side  by  side,  their  fines 
of  diffusion  will  oppose  and  repel  one  another.  Or,  instead  of  the 
uniform  layer  of  salt  solution,  we  may  place  at  a  little  distance 
from  one  another  a  grain  of  salt  and  a  drop  of  blood,  representing 
two  opposite  poles:  and  so  obtain  a  picture  of  a  "bipolar  field" 
of  diffusion.  In  either  case,  we  obtain  results  closely  analogous  to 
the  morphological,  but  really  dynamical,  polarity  of  the  organic 
cell.  But  in  all  probability,  the  dynamical  polarity  or  asymmetry 
of  the  cell  is  a  very  complicated  phenomenon:  for  the  obvious 
reason  that,  in  any  system,  one  asymmetry  will  tend  to  beget 
another.  A  chemical  asymmetry  will  induce  an  inequafity  of 
surface-tension,  which  will  lead  directly  to  a  modification  of  form ; 
the  chemical  asymmetry  may  in  turn  be  due  to  a  process  of 
electrolysis  in  a  polarised  electrical  field;  and  again  the  chemical 
heterogeneity  may  be  intensified  into  a  chemical  polarity,  by  the 
tendency  of  certain  substances  to  seek  a  locus  of  gi^eater  or  less 
surface-energy.  We  need  not  attempt  to  grapple  with  a  subject  so 
compHcated,  and  leading  to  so  many  problems  which  lie  beyond 
the  sphere  of  interest  of  the  morphologist.  But  yet  the  morpho- 
logist,  in  his  study  of  the  cell,  cannot  quite  evade  these  important 
issues;  and  we  shall  return  to  them  again  when  we  have  dealt 
somewhat  with  the  form  of  the  cell,  and  have  taken  account  of 
some  of  its  simpler  phenomena. 

We  are  now  ready,  and  in  some  measure  prepared,  to  study  the 
numerous  and  complex  phenomena  which  accompany  the  division 
of  the  cell,  for  instance  of  the  fertilised  egg.  But  it  is  no  easy  task 
to  epitomise  the  facts  of  the  case,  and  none  the  easier  that  of  late 
new  methods  have  shewn  us  new  things,  and  have  cast  doubt  on 
not  a  little  that  we  have  been  accustomed  to  believe. 

Division  of  the  cell  is  of  necessity  accompanied,  or  preceded,  by 
a  change  from  a  radial  or  monopolar  to  a  definitely  bipolar  sym- 
metry. In  the  hitherto  quiescent  or  apparently  quiescent  cell,  we 
perceive  certain  movements,  which  correspond  precisely  to  what 
must  accompany  and  result  from  a  polarisation  of  forces  within: 
of  forces   which,   whatever  be  their  specific   nature,   are  at  least 


IV]  AND  STRUCTURE  OF  THE  CELL  299 

capable  of  polarisation,  and  of  producing  consequent  attraction  or 
repulsion  between  charged  particles.  The  opposing  forces  which 
are  distributed  in  equilibrium  throughout  the  cell  become  focused 
in  two  "centrosomes*,"  which  may  or  may  not  be  already  visible. 
It  generally  happens  that,  in  the  egg,  one  of  these  centrosomes  is 
near  to  and  the  other  far  from  the  "animal  pole,"  which  is  both 
visibly  and  chemically  different  from  the  other,  and  is  where  the 
more  conspicuous  developmental  changes  will  presently  begin. 

Between  the  two  centrosomes,  in  stained  preparations,  a  spindle- 
shaped  figure  appears  (Fig.  88),  whose  striking  resemblance  to  the 


Fig.  8».     Caryokinetic  ligure  in  a  dividing  cell  (or  blastomere)  of  a  trout  s  egg. 
After  Prenant,  from  a  preparation  by  Prof.  Bouin. 

lines  of  force  made  visible  by  iron-filings  between  the  poles  of  a 
magnet  was  at  once  recognised  by  Hermann  Fol,  in  1873,  when  he 
witnessed  the  phenomenon  for  the  first  timef.  On  the  farther 
side  of  the  centrosomes  are  seen  star-like  figures,  or  "asters,"  in 
which  we  se^m  to  recognise  the  broken  lines  of  force  which  run 
externally  to  those  stronger  lines  which  lie  nearer  to  the  axis  and 
constitute  the  "spindle."  The  lines  of  force  are  rendered  visible, 
or  materialised,  just  as  in  the  experiment  of  the  iron-fihngs,  by  the 
fact  that,  in  the  heterogeneous  substance  of  the  cell,  certain  portions 

*  These  centrosomes  are  the  two  halves  of  a  single  granule,  and  are  said  (by 
Boveri)  to  come  from  the  middle  piece  of  the  original  spermatozoon. 

t  He  did  so  in  the  egg  of  a  medusa  {Geryon),  Jen.  Zeitschr.  vii,  p.  476,  1873. 
Similar  ideas  have  been  expressed  by  Strasbiirger,  Henneguy,  Van  Beneden, 
Errera,  Ziegler,  Gallardo  and  others. 


300  ON  THE  INTERNAL  FORM  [ch. 

of  matter  are  more  "permeable"  to  the  acting  force  than  others, 
become  themselves  polarised  after  the  fashion  of  a  magnetic  or 
"paramagnetic"  body,  arrange  themselves  in  an  orderly  way 
between  the  two  poles  of  the  field  of  force,  seem  to  cling  to  one 
another  as  it  were  in  threads*,  and  are  only  prevented  by  the 
friction  of  the  surrounding  medium  from  approaching  and  con- 
gregating around  the  adjacent  poles. 

As  the  field  of  force  strengthens,  the  more  will  the  lines  of  force 
be  drawn  in  towards  the  interpolar  axis,  and  the  less  evident  will 
be  those  remoter  lines  which  constitute  the  terminal,  or  extrapolar, 
asters:  a  clear  space,  free  from  materialised  fines  of  force,  may 
thus  tend  to  be  set  up  on  either  side  of  the  spindle,  the  so-called 
''Biitschfi  space"  of  the  histologistsl.  On  the  other  hand,  the  lines 
of  force  constituting  the  spindle  will  be  less  concentrated  if  they 
find  a  path  of  less  resistance  at  the  periphery  of  the  ceU :  as  happens 
in  our  experiment  of  the  iron-filings,  when  we  encircle  the  field  of 
force  with  an  iron  ring.  On  this  principle,  the  differences  observed 
between  cells  in  which  the  spindle  is  well  developed  and  the  asters 
small,  and  others  in  which  the  spindle  is  weak  and  the  asters  greatly 
developed,  might  easily  be  explained  by  variations  in  the  potential 
of  the  field,  the  large,  conspicuous  asters  being  correlated  in  turn 
with  a  marked  permeability  of  the  surface  of  the  cell. 

The  visible  field  of  force,  though  often  called  the  "nuclear 
spindle,"  is  formed  outside  of,  but  usually  near  to,  the  nucleus. 


*  Whence  the  name  "mitosis"  (Greek  /ziros,  a  thread),  applied  first  by  Flemming 
to  the  whole  phenomenon.  Kolimann  (Biol.  Centralbl.  ii,  p.  107,  1882)  called  it 
divisio  per  fila,  or  divisio  laqueis  implicata.  Many  of  the  earlier  students,  such  as 
Van  Beneden  (Rech.  sur  la  maturation  de  I'oeuf,  Arch,  de  Biol,  iv,  1883),  and 
Hermann  Fol  (Zur  Lehre  v.  d.  Entstehung  d.  karyokinetischen  Spindel,  Arch.  f. 
mikrosk.  Anat.  x,xxvii,  1891)  thought  they  recognised  actual  muscular  threads, 
drawing  the  nuclear  material  asunder  towards  the  respective  foci  or  poles;  and 
some  such  view  of  Zugkrdfte  was  long  maintained  by  other  writers,  by  Heidenhain 
especially,  by  Boveri,  Flemming,  R.  Hertwig,  Rhumbler,  and  many  more.  In  fact, 
the  existence  of  contractile  threads,  or  the  ascription  to  the  spindle  rather  than  to 
the  poles  or  centrosomes  of  the  active  forces  concerned  in  nuclear  division,  formed 
the  main  tenet  of  all  those  who  declined  to  go  beyond  the  "contractile  properties 
of  protoplasm"  for  an  explanation  of  the  phenomenon  (cf.  J.  W.  Jenkinson, 
Q.J. M.S.  XLViii,  p.  471,  1904.  See  also  J.  Spek's  historical  account  of  the  theories 
of  cell-division.  Arch.  f.  Entw.  Mech.  xliv,  pp.  5-29,  1918). 

t  Cf.  0,  Biitschli,  Ueber  die  kiinstliche  Nachahmung  der  karyokinetischen 
Figur,  Verh.  Med.  Nat.  Ver.  Heidelberg,  v,  pp.  28-41  (1892),  1897. 


IV 


AND  STRUCTURE  OF  THE  CELL 


301 


Let  us  look  a  little  more  closely  into  the  structure  of  this  body, 
and  into  the  changes  which  it  presently  undergoes. 

Within  its  spherical  outHne  (Fig.  89  a),  it  contains  an  ''alveolar" 
meshwork  (often  described,  from  its  appearance  in  optical  section, 
as  a  "reticulum"),  consisting  of  more  sohd  substances  with  more 
fluid  matter  filling  up  the  interalveolar  spaces.  This  phenomenon, 
familiar  to  the  colloid  chemist,  is  what  he  calls  a  "two-phase 
system,"  one  substance  or  "phase"  forming  a  continuum  through 
which  the  other  is  dispersed;   it  is  closely  alhed  to  what  we  call  in 


atr  racHon  -  sphere 
1   ,C€ntro3omes 


Fig.  89  A. 


Fig.  89  B. 


ordinary  language  a.  froth  ot'sl  foam*,  save  that  in  these  latter  the 
disperse  phase  is  represented  by  air.  It  is  a  surface-tension  pheno- 
menon, due  to  the  interaction  of  two  intermixed  fluids  not  very 
different  in  density,  as  they  strive  to  separate.  Of  precisely  the 
same  kind  (as  Biitschli  was  the  first  to  shew)  are  the  minute  alveolar 
networks  which  are  to  be  discerned  in  the  cytoplasm  of  the  cellf, 

*  Froth  and  foam  have  been  much  studied  of  late  years  for  technical  reasons, 
and  other  factors  than  surface-tension  are  foiind  to  be  concerned  in  their  existence 
and  their  stability.  See  (int.  al.)  Freundlich's  Capillarchemie,  and  various  papers 
by  Sasaki,  in  Bull.  Chem.  Soc.  of  Japan,  1936-39. 

t  Biitschli,  Untersuchungen  iiber  mikroskopische  Schdume  und  das  Protoplasma, 
1892;  Untersuchungen  uber  Strukturen,  etc.,  1898;  L.  Rhumbler,  Protoplasma  als 
physikalisches  System,  Ergehn.  d.  Physiologie,  1914;  H.  Giersberg,  Plasmabau 
der  Amoben,  im  Hinblick  auf  die  Wabentheorie,  Arch.  f.  Entw.  Mech.  li,  pp.  150-250, 
1922;   etc. 


302  ON  THE  INTERNAL  FORM  [ch. 

and  which  we  now  know  to  be  not  inherent  in  the  nature  of  proto- 
plasm nor  of  Hving  matter  in  general,  but  to  be  due  to  various 
causes,  natural  as  well  as  artificial*.  The  microscopic  honeycomb 
structure  of  cast  metal  under  various  conditions  of  coohng  is  an 
example  of  similar  surface-tension  phenomena. 

Such  then,  in  briefest  outhne,  is  the  typical  structure  commonly 
ascribed  to  a  cell  when  its  latent  energies  are  about  to  manifest 
themselves  in  the  phenomenon  of  cell-division.  The  account  is 
based  on  observation  not  of  the  hving  cell  but  of  the  dead :  on  the 
assumption,  that  is  to  say,  that  fixed  and  stained  material  gives  a 
true  picture  of  reahty.  But  in  Robert  Chambers's  method  of  micro- 
dissection f,  the  hving  cell  is  manipulated  with  fine  glass  needles 
under  a  high  magnification,  and  shews  us  many  interesting  things. 
Chambers  assures  us  that  the  spindle  fibres  never  make  their 
appearance  as  visible  structures  until  coagulation  has  set  in;  and 
that  astral  rays  are,  or  appear  to  be,  channels  in  which  the  more 
fluid  content  of  the  cell  flows  towards  a  centrosome:|:.  Within  the 
bounds  to  which  we  are  at  present  keeping,  these  things  are  of  no 
great  moment;  for  whether  the  spindle  appear  early  or  late,  it  still 
bears  witness  to  the  fact  that  matter  has  arranged  itself  along 
bipolar  fines  of  force;  and  even  if  the  astral  rays  be  only  streams 
or  currents,  on  lines  of  force  they  still  approximately  he.  Yet  the 
change  from  the  old  story  to  the  new  is  important,  and  may  make 
a  world  of  diff'erence  when  we  attempt  to  define  the  forces  concerned. 
All  our  descriptions,  all  our  interpretations,  are  bound  to  be 
influenced  by  our  conception  of  the  mechanism  before  us;    and  he 

*  Arrhenius,  in  describing  a  typical  colloid  precipitate,  does  so  in  terms  that 
are  very  closely  applicable  to  the  ordinary  microscopic  appearance  of  the  protoplasm 
of  the  cell.  The  precipitate  consists,  he  says,  "en  un  reseau  d'une  substance  solide 
contenant  peu  d'eau,  dans  les  mailles  duquel  est  inclus  un  fluide  contenant  un  peu 
de  colloide  dans  beaucoup  d'eau. . . .  Evidemment  cette  structure  se  forme  a  cause 
de  la  petite  difference  de  poids  specifique  des  deux  phases,  et  de  la  consistance 
gluante  des  particules  separees,  qui  s'attachent  en  forme  de  reseau  "  {Rev.  Scientifique, 
Feb.  1911).  This,  however,  is  far  from  being  the  whole  story:  cf.  (e.g.)  S.  C. 
Bradford,  On  the  theory  of  gels,  Biochem.  Journ.  xvii,  p.  230,  1925;  W.  Seifritz, 
The  alveolar  structure  of  protoplasm,  Protoplasma,  ix,  p.  198,  1930;  and  A.  Frey- 
Wissling,  Submikroskopische  Morphologie  des  Protoplasmas,  Berlin,  1938. 

t  See  R.  Chambers,  An  apparatus. .  .for  the  dissection  and  injection  of  living 
cells,  Anatom.  Record,  xxiv,  19  pp.,  1922. 

X  This  centripetal  flow  of  fluid  was  announced  by  Biitschli  in  his  early  papers, 
and  confirmed  by  Rhumbler,  though  attributed  to  another  cause. 


IV]  AND  STRUCTURE  OF  THE  CELL  303 

who  sees  threads  where  another  sees  channels  is  hkely  to  tell  a 
different  story  about  neighbouring  and  associated  things. 

It  has  also  been  suggested  that  the  spindle  is  somehow  due  to  a  re-arrange- 
ment of  protein  macromolecules  or  micelles ;  that  such  changes  of  orientation 
of  large  colloid  particles  may  be  a  widespread  phenomenon;  and  that  coagu- 
lation itself  is  but  a  polymerisation  of  larger  and  larger  macromolecules*. 

But  here  we  have  touched  the  brink  of  a  subject  so  important  that  we  must 
not  pass  it  by  without  a  word,  and  yet  so  contentious  that  we  must  not  enter 
into  its  details.  The  question  involved  is  simply  whether  the  great  mass  of 
recorded  observations  and  accepted  beliefs  with  regard  to  the  visible  structure 
of  protoplasm  and  of  the  cell  constitute  a  fair  picture  of  the  actual  living  cell, 
or  be  based  on  appearances  which  are  incident  to  death  itself  and  to  the 
artificial  treatment  which  the  microscopist  is  accustomed  to  apply.  The  great 
bulk  of  histological  work  is  done  by  methods  which  involve  the  sudden  killing 
of  the  cell  or  organism  by  strong  reagents,  the  assumption  being  that  death 
is  so  rapid  that  the  visible  phenomena  exhibited  during  life  are  retained  or 
"fixed"  in  our  preparations. 

Hermann  Fol  struck  a  warning  note  full  sixty  years  ago:  "II  importe  a 
I'avenir  de  I'histologie  de  combattre  la  tendance  a  tirer  des  conclusions  des 
images  obtenues  par  des  moyens  artificiels  et  a  leur  donner  une  valeur  intrin- 
seque,  sans  que  ces  images  aient  ete  controlees  sur  le  vivantf."  Fol  was 
thinking  especially  of  cell-membranes  and  the  delimitation  of  cells;  but  still 
more  difficult  and  precarious  is  the  interpretation  of  the  minute  internal  net- 
works, granules,  etc.,  which  represent  the  alleged  structure  of  protoplasm. 
A  colloid  body,  or  colloid  solution,  is  ipso  facto  heterogeneous;  it  has  after 
some  fashion  a  structure  of  its  own.  And  this  structure  chemical  action, 
under  the  microscope,  may  demonstrate,  or  emphasise,  or  alter  and  disguise. 
As  Hardy  put  it,  "It  is  notorious  that  the  various  fixing  reagents  are  co- 
agulants of  organic  colloids,  and  that  the  figure  varies  according  to  the  reagent 
used." 

A  case  in  point  is  that  of  the  vitreous  humour,  to  which  some  histologists 
have  ascribed  a  fairly  complex  structure,  seeing  in  it  a  framework  of  fibres 
with  the  meshes  filled  with  fluid.  But  it  is  really  a  true  gel,  without  any 
structure  in  the  usual  sense  of  the  word.  The  "fibres"  seen  in  ordinary 
microscopic  preparations  are  due  to  the  coagulation  of  micellae  by  the  fixative 
employed.  Under  the  ultra -microscope  the  vitreous  is  optically  empty  to 
begin  with;  then  innumerable  minute  fibrillae  appear  in  the  beam  of  light, 
criss-crossing  one  another.     Soon  these  break  down  into  strings  of  beads,  and 

*  Cf.  J.  D.  Bernal,  on  Molecular  architecture  of  biological  systems,  Proc.  Boy. 
Inst.,  1938;    H.  Staiidinger,  Nature,  Aug.  I,  1939. 

t  H.  Fol,  Becherches  sur  la  fecondation  et  le  commencement  de  VMnogenie  chez 
divers  animaux,  Geneve,  1879,  pp.  241-242.  Cf.  A.  Daleq,  in  Biol.  Beviews,  iii, 
p.  24,  1928:  "II  serait  desirable  de  nous  debarrasser  de  I'idee  que  tout  ce  qu'il 
y  a  d'important  dans  la  cellule  serait  providentiellement  colorable  par  I'hematoxy- 
line,  la  safranine  ou  le  violet  de  gentiane." 


304  ON  THE  INTERNAL  FORM  [ch. 

finally  only  separate  dots  are  seen*.  Other  sources  of  error  arise  from  the 
optical  principles  concerned  in  microscopic  vision;  for  the  diffraction-pattern 
which  we  call  the  "image"  may,  under  certain  circumstances,  be  very  different 
from  the  actual  object  f.  Furthermore,  the  optical  properties  of  living  proto- 
plasm are  especially  complicated  and  imperfectly  known,  as  in  general  those 
of  colloids  may  be  said  to  be;  the  minute  aggregates  of  the  "disperse  phase" 
of  gels  produce  a  scattering  action  on  light,' leading  to  appearances  of  turbidity 
etc.,  with  no  other  or  more  real  basis  J. 

So  it  comes  to  pass  that  some  writers  have  altogether  denied  the  existence 
in  the  living  cell-protoplasm  of  a  network  or  alveolar  "foam";  others  have 
cast  doubts  on  the  main  tenets  of  recent  histology  regarding  nuclear  structure ; 
and  Hardy,  discussing  the  structure  of  certain  gland-cells,  declared  that 
"there  is  no  evidence  that  the  structure  discoverable  in  the  cell-substance  of 
these  cells  after  fixation  has  any  counterpart  in  the  cell  when  living."  "A 
large  part  of  it"  he  went  on  to  say  "is  an  artefact.  The  profound  difference 
in  the  minute  structure  of  a  secretory  cell  of  a  mucous  gland  according  to  the 
reagent  which  is  used  to  fix  it  would,  it  seems  to  me,  almost  suffice  to  establish 
this  statement  in  the  absence  of  other  evidence  §." 

Nevertheless,  histological  study  proceeds,  especially  on  the  part  of  the 
morphologists,  with  but  little  change  in  theory  or  in  method,  in  spite  of  these 
and  many  other  warnings.  That  certain  visible  structures,  nucleus,  vacuoles, 
"attraction-spheres"  or  centrosomes,  etc.,  are  actually  present  in  the  living 
cell  we  know  for  certain;  and  to  this  class  belong  the  majority  of  structures 
with  which  we  are  at  present  concerned.  That  many  other  alleged  structures 
are  artificial  has  also  been  placed  beyond  a  doubt;  but  where  to  draw  the 
dividing  line  we  often  do  not  know. 

The  following  is  a  brief  epitome  of  the  visible  changes  undergone 
by  a  t3rpical  cell,  subsequent  to  the  resting  stage,  leading  up  to  the 
act  of  segmentation,  and  constituting  the  phenomenon  of  mitosis 
or  caryokinetic  division.  In  the  fertilised  egg  of  a  sea-urchin  we 
see  with  almost  diagrammatic  completeness,  in  fixed  and  stained 
specimens,  what  is  set  forth  here||. 

*  W.  S.  Duke-Elder,  Journ.  Physiol,  lxviii,  pp.  1.54-165,  1930;  of.  Baurmann, 
Arch.f.  Ophthalm.  1923,  1926;  etc. 

t  Abbe,  Arch.f.  mikrosk.  Anat.  ix,  p.  413,  1874;  Gesammelte  Ahhandl.  i,  p.  45, 
1904. 

X  Cf.  Rayleigh,  On  the  light  from  the  sky,  Phil.  Mag.  (4)  xli,  p.  107,  1871. 

§  W.  B.  Hardy,  On  the  structure  of  cell  protoplasm,  Journ.  Physiol,  xxiv, 
pp.  158-207,  1889;  also  Hober,  Physikalische  Chemie  der  Zelle  und  der  Gewebe, 
1902;  W.  Berg,  Beitrage  zur  Theorie  der  Fixation,  etc.,  Arch.f.  mikr.  Anat.  LXii,' 
pp.  367-440,  1903,  Cf.  {int.  al.)  Flemming,  Zellsubstanz,  Kern  und  Zelltheilung, 
1882,  p.  51;  etc. 

II  My  description  and  diagrams  (Figs.  89-93)  are  mostly  based  on  those  of 
the  late  Professor  E.  B.  Wilson. 


IV] 


AND  STRUCTURE  OF  THE  CELL 


305 


1.  The  chromatin,  which  to  begin  with  had  been  dimly  seen  as 
granules  on  a  vague  achromatic  reticulum  (Figs.  89,  90) — perhaps  no 
more  than  an  histological  artefact — concentrates  to  form  a  skein  or 
spireme,  often  looked  on  as  a  continuous  thread,  but  perhaps 
discontinuous  or  fragmented  from  the  first.  It,  or  its  several 
fragments,  will  presently  spht  asunder;  for  it  is  essentially  double, 
and  may  even  be  seen  as  a  double  thread,  or  pair  of  chromatids,  from 
an  early  stage.  The  chromosomss  are  portions  of  this  double  thread, 
which  shorten  down  to  form  httle  rods,'  straight  or  curved,  often 


chromoaome* 


Fig.  90  A. 


Fig.  90  B. 


bent  into  a  V,  sometimes  ovoid,  round  or  even  annular,  and  which 
in  the  living  cell  are  frequently  seen  in  active,  writhing  movement, 
*"hke  eels  in  a  box"*;  they  keep  apart  from  one  another,  as  by 
some  repulsion,  and  tend  to  move  outward  towards  the  nuclear 
membrane.  Certain  deeply  staining  masses,  the  nucleoh,  may  be 
present  in  the  resting  nucleus,  but  take  no  part  (at  least  as  a  rule) 
in  the  formation  of  the  chromosomes;  they  are  either  cast  out  of 
the  nucleus  and  dissolved  in  the  cytoplasm,  or  else  fade  away  in  situ. 

*  T.  S.  Strangeways,  Proc.  E.S.  (B),  xciv,  p.  139,  1922.  The  tendency  of  the 
chromatin  to  form  spirals,  large  or  small,  while  the  nucleus  is  issuing  from  its 
resting-stage,  is  very  remarkable.  The  tensions  to  which  it  is  due  may  be  overcome, 
and  the  chromosomes  made  to  uncoil,  by  treatment  with  ammonia  or  acetic  acid 
vapour.  See  Y.  Kuwada,  Botan.  Mag.  Tokyo,  xlvi,  p.  307,  1932;  and  C.  D. 
Darlington,  Mechanical  aspects  of  nuclear  division,  Sci.  Journ.  B.  Coll,  of  Sci. 
TV,  p.  94,  1934. 


306  ON  THE  INTERNAL  FORM  [ch. 

But  this  rule  does  not  always  hold;  for  they  persist  in  many 
protozoa,  and  now  and  then  the  nucleolus  remains  and  becomes 
itself  a  chromosome,  as  in  the  spermogonia  of  certain  insects. 

2.  Meanwhile  a  certain  deeply  staining  granule  (here  extra- 
nuclear),  known  as  the  centrosome*,  has  divided  into  two.  It  is  all 
but  universally  visible,  save  in  the  higher  plants;  perhaps  less  stress 
is  laid  on  it  than  at  one  time,  but  Bovery  called  i-t  the  "dynamic 
centre"  of  the  cellf.  The  two  resulting  granules  travel  to 
opposite  poles  Df  the  nucleus,  and  there  eacii  becomes  surrounded 
by  a  starhke  figure,  the  aster,  of  which  we  have  sptken  already; 
immediately  around  the  centrosome  is  a  clear  space,  the  centro- 
sphere.  Between  the  two  centrosomes,  or  the  two  asters,  stretches 
the  spindle.  It  lies  in  the  long  axis,  if  there  be  one,  of  the  cell,  a 
rule  laid  down  nearly  sixty  years  ago,  and  still  remembered  as 
"Hertwig's  Law"  J;  but  the  rule  is  as  much  and  no  more  than  to 
say  that  the  spindle  sets  in  the  direction  of  least  resistance.  Where 
the  egg  is  laden  with  food-yolk,  as  often  happens,  the  latter  is 
heavier  than  the  cytoplasm;  and  gravity,  by  orienting  the  egg 
itself,  thus  influences,  though  only  indirectly,  the  first  planes  of 
segmentation  §. 

3.  The  definite  nuclear  outhne  is  soon  lost;  for  the  chemical 
"phase-difference"  between  nucleus  and  cytoplasm  has  broken 
down,  and  where  the  nucleus  was,  the  chromosomes  now  he  (Figs. 
90,  91).  The  lines  of  the  spindle  become  visible,  the  chromosomes 
arrange  themselves  midway  between  its  poles,  to  form  the  equatorial 
plate,  and  are  spaced  out  evenly  around  the  central  spindle,  again 
a  simple  result  of  mutual  repulsion. 

4.  Each  chromosome  separates  longitudinally  into  two|| :  usually 
at  this  stage — but  it  is  to  be  noted  that  the  spHtting  may  have  taken 
place  as  early  as  the  spireme  stage  (Fig.  92). 

*  The  centrosome  has  a  curious  history  of  its  own,  none  too  well  ascertained. 
The  ovum  has  a  centrosome,  and  in  self- fertilised  eggs  this  is  retained;  but  when 
a  sperm-cell  enters  the  egg  the  original  centrosome  degenerates,  and  its  place  is 
taken  by  the  "middle-piece"  of  the  spermatozoon, 

f  The  stages  1,  2,  5  and  6  are  called  by  embryologista  the  prophase,  metaphase, 
anaphase  and  telophase. 

X  C.  Hertwig,  Jenaische  Ztschr.  xviii,  1884. 

§  See  James  Gray,  The  effect  of  gravity  on  the  eggs  of  Echinus,  Jl.  Exp.  Zool.  v, 
pp.  102-11,  1927. 

II  A  fundamental  fact,  first  seen  by  Flemming  in  1880. 


IV] 


AND  STRUCTURE  OF  THE  CELL 


307 


5.  The  halves  of  the  spht  chromosomes  now  separate  from  and 
apparently  repel  one  another,  travelUng  in  opposite  directions 
towards  the  two  poles*  (Fig.  92  b),  for  all  the  world  as  though  they 
were  being  pulled  asunder  by  actual  threads. 


Fig.  91  A. 


Fig.  91  B. 


central  spindle 

mantle 'fibres 


split  chromosome* 


Fig.  92  A. 


Fig.  92  B. 


6.     Presently  the  spindle  itself  changes  shape,  lengthens  and  con- 
tracts, and  seems  as  it  were  to  push  the  two  groups  of  daughter- 


*  Cf.  K.  Belar,  Beitrage  zur  Causalanalyse  der  Mitose,  Ztschr.  f.  Zellforschung, 
X,  pp.  73-124,  1929. 


308 


ON  THE  INTERNAL  FORM 


[CH. 


chromosomes  into  their  new  places*  (Figs.  92,  93);  and  its  chromo- 
somes form  once  more  an  alveolar  reticulum  and  may  occasionally 
form  another  spireme  at  this  stage.  A  boundary-surface,  or  at  least  a 
recognisable  phase-difference,  now  develops  round  each  reconstructed 
nuclear  mass,  and  the  spindle  disappears  (Fig.  93  b).  The  centrosome 
remains,  as  a  rule,  outside  the  nucleus. 

7.  On  the  central  spindle,  in  the  position  of  the  equatorial  plate, 
a  "cell-plate,"  consisting  of  deeply  staining  thickenings,  has  made 
its  appearance  during  the  migration  of  the  chromosomes.  This  cell- 
plate  is  more  conspicuous  in  plant-cells. 


otfrottron  spncft 


,d>iopptQr,ng  spmifte 


Htcon^rucffd  dough/trnuei^t 


Fig.  93  B. 


8.  Meanwhile  a  constriction  has  appeared  in  the  cytoplasm,  and 
the  cell  divides  through  the  equatorial  plane.  In  plant-cells  the 
line  of  this  division  is  foreshadowed  by  the  "cell-plate,"  which 
extends  from  the  spindle  across  the  entire  cell,  and  spHts  into  two 
layers,  between  which  appears  the  membrane  by  which  the  daughter- 
cells  are  cleft  asunder.  In  animal  cells  the  cell-plate  does  not  attain 
such  dimensions,  and  no  cell-wall  is  formed. 

The  whole  process  takes  from  half-an-hour  to  an  hour;  and  this 
extreme  slowness  is  not.  the  least  remarkable  part  of  the  pheno- 
menon, from  a  physical  point  of  view.     The  two  halves  of  the 

*  The  spindle  has  no  actual  threads  or  fibres,  for  Robert  Chambers's  micro- 
needles pass  freely  through  it  without  disturbing  the  chromosomes:  nor  is  it 
visible  at  all  in  living  cells  in  vitro.  It  seems  to  be  due  to  partial  gelation  of  the 
cytoplasm,  under  conditions  which,  whether  they  be  mechanical  or  chemical,  are 
not  easy  to  understand. 


IV]  AND  STRUCTURE  OF  THE  CELL  309 

dividing  centrosome,  while  moving  apart,  take  some  twenty  minutes 
to  travel  a  distance  of  20 /x,  or  at  the  rate,  say,  of  two  years  to  a 
yard.  It  is  a  question  of  inertia,  and  the  inertia  of  the  system  must 
be  very  large. 

The  beautiful  technique  of  cell-culture  in  vitro  has  of  late  years 
let  this  whole  succession  of  phenomena,  once  only  to  be  deduced 
from  sections,  be  easily  followed  as  it  proceeds  within  the  living 
tissue  or  cell.  The  vivid  accounts  which  have  been  given  of  this 
spectacle  add  little  to  the  older  account  as  we  have  related  it: 
save  that,  when  the  equatorial  constriction  begins  and  the  halves 
of  the  split  chromosomes  drift  apart,  the  protoplasm  begins  to  show 
a  curious  and  even  violent  activity.  The  cytoplasm  is  thrust  in 
and  out  in  bulging  pustules  or  "balloons";  and  the  granules  and 
fat-globules  stream  in  and  out  as  the  pustules  rise  and  fall  away. 
At  length  the  turmoil  dies  down;  and  now  each  half  of  the  cell 
(not  an  ovum  but  a  tissue-cell  or  "fibroplast")  pushes  out  large 
pseudopodia,  flattens  into  an  amoeboid  phase,  the  connecting  thread 
of  protoplasm  snaps  in  the  divided  cell,  and  the  daughter-cells  fall 
apart  and  crawl  away.  The  two  groups  of  chromosomes,  on  reaching 
the  poles  of  the  spindle,  turn  into  bunches  of  short  thick  rods;  these 
grow  diffuse,  and  form  a  network  of  chromatin  within  a  nucleus; 
and  at  last  the  chromosomes,  having  lost  their  identity,  disappear 
entirely,  and  two  or  more  nucleoH  are  all  that  is  to  be  seen  within 
the  cell. 

The  whole,  or  very  nearly  the  whole,  of  these  nuclear  phenomena 
may  be  brought  into  relation  with  some  such  polarisation  of  forces 
in  the  cell  as  a  whole  as  is  indicated  by  the  "spindle"  and  "asters" 
of  which  we  have  already  spoken:  certain  particular  phenomena, 
directly  attributable  to  surface-tension  and  diffusion,  taking  place 
in  more  or  less  obvious  and  inevitable  dependence  upon  the  polar 
system.  At  the  same  time,  in  attempting  to  explain  the  phenomena, 
we  cannot  say  too  clearly,  or  too  often,  that  all  that  we  are  meanwhile 
justified  in  doing  is  to  try  to  shew  that  such  and  such  actions  He 
within  the  range  of  known  physical  actions  and  phenomena,  or  that 
known  physical  phenomena  produce  effects  similar  to  them.  We 
feel  that  the  whole  phenomenon  is  iiot  sui  generis,  but  is  some- 
how or  other  capable  of  being  referred  to  dynamical  laws,  and  to 


310  ON  THE  INTERNAL  FORM  [ch. 

the  general  principles  of  physical  science.  But  when  we  speak  of 
some  particular  force  or  mode  of  action,  using  it  as  an  illustrative 
hypothesis,  we  stop  far  short  of  the  implication  that  this  or  that 
force  is  necessarily  the  very  one  which  is  actually  at  work  within 
the  living  cell;  and  certainly  we  need  not  attempt  the  formidable 
task  of  trying  to  reconcile,  or  to  choose  between,  the  various 
hypotheses  which  have  already  been  enunciated,  or  the  several 
assumptions  on  which  they  depend. 

Many  other  things  happen  within  the  cell,  especiall)^  in  the  germ- 
cell  both  before  and  after  fertilisation.  They  also  have  a  physical 
element,  or  a  mechanical  aspect,  like  the  phenomena  of  cell- 
division  which  we  are  speaking  of;  but  the  narrow  bounds  to  which 
we  are  keeping  hold  difficulties  enough*. 

Any  region  of  space  within  which  action  is  manifested  is  a  field 
of  force;  and  a  simple  example  is  a  bipolar  field,  in  which  the 
action  is  symmetrical  with  reference  to  the  fine  joining  two  points, 
or  poles,  and  with  reference  also  to  the  "equatorial"  plane  equi- 
distant from  both.  We  have  such  a  field  of  force  in  the  neigh- 
bourhood of  the  centrosome  of  the  ripe  cell  or  ovum,  when  it  is 
about  to  divide;  and  by  the  time  the  centrosome  has  divided,  the 
field  is  definitely  a  bipolar  one. 

The  quality  of  a  medium  filling  the  field  of  force  may  be  uniform, 
or  it  may  vary  from  point  to  point.  In  particular,  it  may  depend 
upon  the  magnitude  of  the  field;  and  the  quality  of  one  medium 
may  differ  from  that  of  another.  Such  variation  of  quality,  within 
one  medium,  or  from  one  medium  to  another,  is  capable  of  diagram- 
matic representation  by  a  variation  of  the  direction  or  the  strength 
of  the  field  (other  conditions  being  the  same)  from  the  state 
manifested  in  some  uniform  medium  taken  as  a  standard.  The 
medium  is  said  to  be  permeable  to  the  force,  in  greater  or  less  degree 
than  the  standard  medium,  according  as  the  variation  of  the  density 
of  the  lines  of  force  from  the  standard  case,  under  otherwise  identical 
conditions,  is  in  excess  or  defect.  A  body  placed  in  the  medium  will 
tend  to  move  towards  regions  of  greater  or  less  force  according  as  its 

*  Cf.  C.  D.  Darlington,  JieretU  Advances  in  Cytology,  1932,  and  other  well-known 
works. 


IV]  AND  STRUCTURE  OF  THE  CELL  311 

penneability  is  greater  or  less  than  that  of  the  surrounding  medium'^. 
In  the  common  experiment  of  placing  iron-filings  between  the  two 
poles  of  a  magnetic  field,  the  filings  have  a  very  high  permeability; 
and  not  only  do  they  themselves  become  polarised  so  as  to  attract 
one  another,  but  they  tend  to  be  attracted  from  the  weaker  to  the 
stronger  parts  of  the  field,  and  as  we  have  seen,  they  would  soon 
gather  together  around  the  nearest  pole  were  it  not  for  friction 
or  some  other  resistance.  But  if  we  "repeat  the  same  experiment 
with  such  a  metal  as  bismuth,  which  is  very  little  permeable  to  the 
magnetic  force,  then  the  conditions  are  reversed,  and  the  particles, 
being  repelled  from  the  stronger  to  the  weaker  parts  of  the  field, 
tend  to  take  up  their  position  as  far  from  the  poles  as  possible. 
The  particles  have  become  polarised,  but  in  a  sense  opposite  to  that 
of  the  surrounding,  or  adjacent,  field. 

Now,  in  the  field  of  force  whose  opposite  poles  are  marked  by 
the  centrosomes,  we  may  imagine  the  nucleus  to  act  as  a  more  or 
less  permeable  body,  as  a  body  more  permeable  than  the  surrounding 
medium,  that  is  to  say  the  "  cytoplasm  "  of  the  cell.  It  is  accordingly 
attracted  by,  and  drawn  into,  the  field  of  force,  and  tries,  as  it 
were,  to  set  itself  between  the  poles  and  as  far  as  possible  from  both 
of  them.  In  other  words,'  the  centrosome-foci  will  be  apparently 
drawn  over  its  surface,  until  the  nucleus  as  a  whole  is  involved 
within  the  field  of  force  which  is  visibly  marked  out  by  the  "spindle" 
(Fig.  90  b). 

If  the  field  of  force  be  electrical,  or  act  in  a  fashion  analogous 
to  an  electrical  field,  the  charged  nucleus  will  have  its  surface- 
tensions  diminished  f:  with  the  double  result  that  the  inner  alveolar 
mesh  work  will  be  broken  up  (par.  1),  and  that  the  spherical 
boundary  of  tKe  whole  nucleus  will  disappear  (par.  2).  The  break- 
up of  the  alveoli  (by  thinning  and  rupture  of  their  partition  walls) 


*  If  the  word  penneability  be  deemed  too  directly  suggestive  of  the  phenomena 
of  magnetism,  we  may  replace  it  by  the  more  general  term  of  specific  iyidiictive 
capacity.  This  would  cover  the  particular  case,  which  is  by  no  means  an  improbable 
one,  of  our  phenomena  being  due  to  a  "surface  charge"  borne  by  the  nucleus 
itself  and  also  by  the  chromosomes:  this  surface  charge  being  in  turn  the  result 
of  a  difference  in  inductive  capacity  between  the  body  or  particle  and  its  surrounding 
medium. 

t  On  the  effect  of  electrical  influences  in  altering  the  surface-tensions  of  the 
colloid  particles,  see  Bredig,  Anorganische  Fermente,  pp.  15,  16,  1901. 


312  ON  THE  INTERNAL  FORM  [ch. 

leads  to  the  formation  of  a  net,  and  the  further  break-up  of  the  net 
may  lead  to  the  unravelling  of  a  thread  or  "spireme". 

Here  there  comes  into  play  a  fundamental  principle  which,  in 
so  far  as  we  require  to  understand  it,  can  be  explained  in  simple 
words.  The  eifect  (and  we  might  even  say  the  object)  of  drawing 
the  more  permeable  body  in  between  the  poles  is  to  obtain  an 
"easier  path"  by  which  the  Hues  of  force  may  travel;  but  it  is 
obvious  that  a  longer  route  through  the  more  permeable  body  may 
at  length  be  found  less  advantageous  than  a  shorter  route  through 
the  less  permeable  medium.  That  is  to  say,  the  more  permeable 
body  will  only  tend  to  be  drawn  into  the  field  of  force  until  a  point 
is  reached  where  (so  to  speak)  the  way  round  and  the  way  through 
are  equally  advantageous.  We  should  accordingly  expect  that  (on 
our  hjrpothesis)  there  would  be  found  cases  in  which  the  nucleus 
was  wholly,  and  others  in  which  it  was  only  partially,  and  in  greater 
or  less  degree,  drawn  in  to  the  field  between  the  centrosomes.  This 
is  precisely  what  is  found  to  occur  in  actual  fact.  Figs.  90  a  and  b 
represent  two  so-called  "types,"  of  a  phase  which  follows  that 
represented  in  Fig.  89.  According  to  the  usual  descriptions  we  are 
told  that,  in  such  a  case  as  Fig.  90b,  the  "primary  spindle" 
disappears*  and  the  centrosomes  diverge  to  opposite  poles  of  the 
nucleus;  such  a  condition  being  found  in  many  plant-cells,  and  in 
the  cleavage-stages  of  many  eggs.  In  Fig.  90  a,  on  the  other  hand, 
the  primary  spindle  persists,  and  subsequently  comes  to  form  the 
main  or  "central"  spindle;  while  at  the  same  time  we  see  the 
fading  away  of  the  nuclear  membrane,  the  breaking  up  of  the 
spireme  into  separate  chromosomes,  and  an  ingrowth  into  the  nu- 
clear area  of  the  "astral  rays" — all  as  in  Fig.  91  a,  which  represents 
the  next  succeeding  phase  of  Fig.  90  b.  This  condition,  of  Fig.  91  a, 
occurs  in  a  variety  of  cases;  it  is  well  seen  in  the  epidermal  cells 
of  the  salamander,  and  is  also  on  the  whole  characteristic  of  the 
mode  of  formation  of  the  "polar  bodies t."  It  is  clear  and  obv;ous 
that  the  two  "types"  correspond  to  mere  differences  of  degree, 

*  The  spindle  is  potentially  there,  even  though  (as  Chambers  assures  us)  it  only 
becomes  visible  after  post-mortem  coagulation.  It  is  also  said  to  become  visible 
under  crossed  nicols:  W.  J.  Schmidt,  Biodynamica,  xxii,  1936. 

t  These  v/ere  first  observed  in  the  egg  of  a  pond-snail  (Limnaea)  by  B.  Dumortier, 
Mim.  sur  Vemhryoginie  des  mollusques,  Bruxelles,  1837. 


IV]  AND  STRUCTURE  OF  THE  CELL  313 

and  are  such  as  would  naturally  be  brought  about  by  differences 
in  the  relative  permeabilities  of  the  nuclear  mass  and  of  the 
surrounding  cytoplasm,  or  even  by  differences  in  the  magnitude  of 
the  former  body. 

But  now  an  important  change  takes  place,  or  rather  an  important 
difference  appears;  for,  whereas  the  nucleus  as  a  whole  tended  to 
be  drawn  in  to  the  stronger  parts  of  the  field,  when  it  comes  to  break 
up  we  find,  on  the  contrary,  that  its  contained  spireme-thread  or 
separate  chromosomes  tend  to  be  repelled  to  the  weaker  parts. 
Whatever  this  difference  may  be  due  to — whether,  for  instance,  to 
actual  differences  of  permeability,  or  possibly  to  differences  in 
"surface-charge"  or  to  other  causes — the  fact  is  that  the  chromatin 
substance  now  behaves  after  the  fashion  of  a  "diamagnetic''  body, 
and  is  repelled  from  the  stronger  to  the  weaker  parts  of  the  field. 
In  other  words,  its  particles,  lying  in  the  inter-polar  field,  tend  to 
travel  towards  the  equatorial  plane  thereof  (Figs.  91,  92),  and 
further  tend  to  move  outwards  towards  the  periphery  of  that  plane, 
towards  what  the  histologist  calls  the  "mantle-fibres,"  or  outermost 
of  the  lines  of  force  of  which  the  spindle  is  made  up  (par.  5,  Fig.  91  b). 
And  if  this  comparatively  non-permeable  chromatin  substance  come 
to  consist  of  separate  portions,  more  or  less  elongated  in  form, 
these  portions,  or  separate  "chromosomes,"  will  adjust  themselves 
longitudinally,  in  a  peripheral  equatorial  circle  (Figs.  92  a,  b).  This 
is  precisely  what  actually  takes  place.  Moreover,  before  the  breaking 
up  of  the  nucleus,  long  before  the  chromatin  material  has  broken 
up  into  separate  chromosomes,  and  at  the  v^ry  time  when  it  is 
being  fashioned  into  a  "spireme,"  this  body  already  lies  in  a  polar 
field,  and  must  already  have  a  tendency  to  set  itself  in  the  equatorial 
plane  thereof.  But  the  long,  continuous  spireme  thread  is  unable, 
so  long  as  the  nucleus  retains  its  spherical  boundary  wall,  to  adjust 
itself  in  a  simple  equatorial  annulus;  in  striving  to  do  so,  it  must 
tend  to  coil  and  "kink"  itself,  and  in  so  doing  (if  all  this  be  so), 
it  must  t^nd  to  assume  the  characteristic  convolutions  of  the 
"spireme." 

After  the  spireme  has  broken  up  into  separate  chromosomes, 
these  bodies  come  to  rest  in  the  equatorial  plane,  somewhere  near 
its  periphery ;  and  here  they  tend  to  set  themselves  in  a  symmetrical 
arrangement  (Fig.  94),  such  as  makes  for  still  better  equihbrium. 


314 


ON  THE  INTERNAL  FORM 


[CH. 


The  particles  ir^y  be  rounded  or  linear,  straight  or  bent,  sometimes 
annular;  they  may  be  all  alike,  or  one  or  more  may  differ  from 
the  rest.  Lying  as  they  do  in  a  semi-fluid  medium,  and  subject 
(doubtless)  to  some  symmetrical  play  of  forces,  it  is  not  to  be 
wondered  at  that  they  arrange  themselves  in  a  symmetrical  con- 
figuration; and  the  field  of  force  seems  simple  enough  to  let  us 
predict,  to  some  extent,  the  symmetries  open  to  them.  We  do  not 
know,  we  cannot  safely  surmise,  the  nature  of  the  forces  involved. 
In  discussing  Brauer's  observations  on  the  sphtting  of  the  chromatic 
filament,  and  on  the  symmetrical  arrangement  of  the  separate 
granules,  in  Ascaris  megalocephala,  LiUie*  remarks:  "This  behaviour 


Fig.  94.     Chromosomes,  undergoing  splitting  and  separation. 
After  Hatsehek  and  Flemming,  diagrammatised. 


is  strongly  suggestive  of  the  division  of  a  colloidal  particle  under 
the  influence  of  its  surface  electrical  charge,  and  of  the  effects  of 
mutual  repulsion  in  keeping  the  products  of  division  apart."  It  is 
probable  that  surface-tensions  between  the  particles  and  the  sur- 
rounding protoplasm  would  bring  about  an  identical  result,  and 
would  sufficiently  account  for  the  obvious,  and  at  first  sight  very 
curious  symmetry.  If  we  float  a  couple  of  matches  in  water,  we 
know  that  they  tend  to  approach  one  another  till  they  He  close 
together,  side  by  side;  and  if  we  lay  upon  a  smooth  wet  plate 
four  matches,  half  broken  across,  a  similar  attraction  brings  the 
four  matches  together  in  the  form  of  a  symmetrical  cross.  Whether 
one  of  these,  or  yet  another,  be  the  explanation  of  the  phenomenon, 


*  R.  S.  Lillie,  Conditions  determining  the  disposition  of  the  chromatic  filaments, 
etc.,  in  mitosis;   Biol.  Bulletin,  viii,  1905. 


IV]  AND  STRUCTURE  OF  THE  CELL  315 

it  is  at  least  plain  that  by  some  physical  cause,  some  mutual 
attraotion  or  common  repulsion  of  the  particles,  we  must  seek  to 
account  for  the  symmetry  of  the  so-called  "tetrads,"  and  other 
more  or  less  familiar  configurations.  The  remarkable  annular 
chromosomes,  shewn  in  Fig.  95,  can  be  closely  imitated  by  loops 
of  thread  upon  a  soapy  film,  when  the  film  within  the  annulus  is 
broken  or  its  tension  reduced ;  the  balance  of  forces  is  here  a  simple 
one,  between  the  uniform  capillary  tension  which  tends  to  widen  out 
the  ring  and  the  uniform  cohesion  of  its  particles  which  keeps  it 
together. 

We  may  find  other  cases,  at  once  simpler  and  more  varied,  where 
the  chromosomes  are  bodies  of  rounded  form  and  more  or  less 


Fig.  95.     Annular  chromosomes,  formed  in  the  spermatogenesis  of  the 
mole-cric'iict.     From  Wilson,  after  Vom  Rath. 

uniform  size.  These  also  find  their  way  to.  an  equatorial  plate; 
we  gather  (and  Lamb  assures  us)  that  they  are  repelled  from  the 
centrosomes.  They  may  go  near  the  equatorial  periphery,  but  they 
are  not  driven  there;  and  we  infer  that  some  bond  of  mutual 
attraction  holds  them  together.  If  they  be  free  to  move  in  a  fluid 
medium,  subject  both  to  some  common  repulsion  and  some  mutual 
attraction,  then  their  circumstances  are  much  like  those  of  Mayer's 
well-known  experiment  of  the  floating  magnets.  A  number  of 
magnetised  needles  stuck  in  corks,  all  with  like  poles  upwards,  are 
set  afloat  in  a  basin;  they  repel  one  another,  and  scatter  away  to 
the  sides.  But  bring  a  strong  magnet  (of  unlike  pole)  overhead, 
and  the  little  magnets  gather  in  under  its  common  attraction,  while 
still  keeping  asunder  through  their  own  mutual  repulsion.  The 
symmetry  of  forces  leads  to  a  symmetrical  configuration,  which  is 


316  ON  THE  INTERNAL  FORM  [ch. 

the  mathematical  expression  of  a  physical  equiUbrium — and  is  the 
not  too  remote  counterpart  of  the  arrangement  of  the  electrons  in 
an  atom.  Be  that  as  it  may,  it  is  found  that  a  group  of  three, 
four  or  five  Httle  magnets  arrange  themselves  at  the  corners  of  an 
equilateral  triangle,  square  or  pentagon;  but  a  sixth  passes  within 
the  ring,  and  comes  to  rest  in  the  centre  of  symmetry  of  the 
pentagon.  If  there  be  seven  magnets,  six  form  the  ring,  and  the 
seventh  occupies  the  centre ;  if  there  be  ten,  there  is  a  ring  of  eight 
and  two  within  it ;   and  so  on,  as  follows  * : 


Number  of  magnets 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

Do.  in  outer  ring 
Do.  in  inner  ring 

5 
0 

5 

1 

6 

1 

7 
1 

8 

1 

8 
2 

8 
3 

9 
3 

10 
3 

10 

4 

10 
5 

11 
5 

When  we  choose  from  the  published  figures  cases  where  the 
chromosomes  are  as  nearly  as  possible  alike  in  size  and  form — the 
condition  necessary  for  our  parallel  to  hold — then,  as  LilHe  pre- 
dicted and  as  Doncaster  and  Graham  Cannon  have  shewn,  their 
congruent  arrangement  agrees,  even  to  a  surprising  degree,  with 
what  we  are  led  to  expect  by  theory  and  analogy  (Fig.  96). 

The  break-up  of  the  nucleus,  already  referred  to  and  ascribed 
to  a  diminution  of  its  surface-tension,  is  accompanied  by  certain 
diffusion  phenomena  which  are  sometimes  visible  to  the  eye;  and 
we  are  reminded  of  Lord  Kelvin's  view  that  diffusion  is  implicitly 
associated  with  surface-tension  changes,  of  which  the  first  step  is 
a  minute  puckering  of  the  surface-skin,  a  sort  of  interdigitation  with 
the  surrounding  medium.  For  instance,  Schewiakoff  has  observed 
in  Euglyphaf  that,  just  before  the  break-up  of  the  nucleus,  a  system 
of  rays  appears,  concentred  about  it,  but  having  nothing  to  do  with 
the  polar  asters:  and  during  the  existence  of  this  striation  the 
nucleus  enlarges  very  considerably,  evidently  by  imbibition  of  fluid 
from  the  surrounding  protoplasm.  In  short,  diffusion  is  at  work, 
hand  in  hand  with,  and  as  it  were  in  opposition  to,  the  surface- 
tensions  which  define  the  nucleus.  By  diffusion,  hand  in  hand  with 
surface-tension,  the  alveoli  of  the  nuclear  meshwork  are  formed, 
enlarged  and  finally  ruptured:   diffusion  sets  up   the   movements 

*  H.  Graham  Cannon,  On  the  nature  of  the  centrosomal  force,  Journ.  Genetics, 
XIII,  p.  55,  1923. 

t  Schewiakoff,  Ueber  die  karyokinetische  Kerntheilung  der  Euglypha  alveolata, 
Morph.  Jahrb.  xiii,  pp.  193-258,  1888  (see  p.  216). 


IV]  AND  STRUCTURE  OF  THE  CELL  317 

which  give  rise  to  the  appearance  of  rays,  or  striae,  around  the 
nucleus:  and  through  increasing  diffusion  and  weakening  surface- 
tension  the  rounded  outHne  of  the  nucleus  finally  disappears. 

As  we  study  these  manifold  phenomena  in  the  individual  cases 
of  particular  plants  and  animals,  we  recognise  a  close  identity  of 
type  coupled  with  almost  endless  variation  of  specific  detail;    and 


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Fig.  96.  Various  numbers  of  chromosomes  in  the  equatorial  plate:    the  ring- 
diagrams  give  the  arrangements  predicted  by  theory.     From  Graham  Cannon. 

in  particular,  the  order  of  succession  in  which  certain  of  the  pheno- 
mena occur  is  variable  and  irregular.  The  precise  order  of  the 
phenomena,  the  tiiiie  of  longitudinal  and  of  transverse  fission  of 
the  chromatin  thread,  of  the  break-up  of  the  nuclear  wall,  and  so 
forth,  will  depend  upon  various  minor  contingencies  and  ''inter- 
ferences." And  it  is  worthy  of  particular  note  that  these  variations 
in  the  order  of  events  and  in  other  subordinate  details,   while 


318  ON  THE  INTERNAL  FORM  [ch. 

doubtless  attributable  to  specific  physical  conditions,  would  seem 
to  be  without  any  obvious  classificatory  meaning  or  other  biological 
significance. 

So  far  as  we  have  now  gone,  there  is  no  great  difficulty  in  pointing 
to  simple  and  familiar  examples  of  a  field  of  force  which  are 
similar,  or  comparable,  to  the  phenomena  which  we  witness  within 
the  cell.  But  among  these  latter  phenomena  there  are  others  for 
which  it  is  not  so  easy  to  suggest,  in  accordance  with  known  laws, 
a  simple  mode  of  physical  causation.  It  is  not  at  once  obvious 
how,  in  any  system  of  symmetrical  forces,  the  chromosomes,  which 
had  at  first  been  apparently  repelled  from  the  poles  towards  the 
equatorial  plane,  should  then  be  spht  asunder,  and  should  presently 
be  attracted  in  opposite  directions,  some  to  one  pole  and  some  to 
the  other.  Remembering  that  it  is  not  our  purpose  to  assert  that 
some  one  particular  mode  of  action  is  at  work,  but  merely  to  shew 
that  there  do  exist  physical  forces,  or  distributions  of  force,  which 
are  capable  of  producing  the  required  result,  I  give  the  following 
suggestive  hypothesis,  which  I  owe  to  my  colleague  Professor  W. 
Peddie. 

As  we  have  begun  by  supposing  that  the  nuclear  or  chromosomal 
matter  differs  in  permeability  from  the  medium,  that  is  to  say  the 
cytoplasm,  in  which  it  hes,  let  us  now  make  the  further  assumption 
that  its  permeabihty  is  variable,  and  depends  upon  the  strength  of 
the  field. 

In  Fig.  97,  we  have  a  field  of  force  (representing  our  cell),  con- 
sisting of  a  homogeneous  medium,  and  including  two  opposite 
poles :  lines  of  force  are  indicated  by  full  lines,  and  loci  of  constant 
magnitude  of  force  are  shewn  by  dotted  lines,  these  latter  being  what 
are  known  as  Cayley's  equipotential  curves*. 

Let  us  now  consider  a  body  whose  permeabihty  (/a)  depends  on 
the  strength  of  the  field  F.  At  two  field -strengths,  such  as  F^,  F^, 
let  the  permeability  of  the  body  be  equal  to  that  of  the  medium, 
and  let  the  curved  line  in  Fig.  98  represent  generally  its  permeabihty 
at  other  field-strengths;  and  let  the  outer  and  inner  dotted  curves 
in  Fig.  97  represent  respectively  the  loci  of  the  field-strengths  F^, 

*  Phil.  Trans,  xiv,  p.  142,  1857.  Cf.  also  F.  G.  Teixeira,  TraiU  des  Courhes, 
I,  p.  372,  Coimbra'.  1908. 


IV]  AND  STRUCTURE  OF  THE  CELL  319 

and  Fa.  The  body  if  it  be  placed  in  the  medium  within  either 
branch  of  the  inner  curve,  or  outside  the  outer  curve,  will  tend  to 
move  into  the  neighbourhood  of  the  adjacent  pole.     If  it  be  placed 


Fb 


Fig.  97. 


Fig.  98. 


in  the  region  intermediate  to  the  two  dotted  curves,  it  will  tend  to 
move  towards  regions  of  weaker  field-strength. 

The  locus  Fly  is  therefore  a  locus  of  stable  position,  towards  which 
the  body  tends  to  move ;  the  locus  F^  is  a  locus  of  unstable  position, 
from  which  it  tends  to  move.     If  the  body  were  placed  across  F^^, 


320  ON  THE  INTERNAL  FORM  [ch. 

it  might  be  torn  asunder  into  two  portions,  the  split  coinciding 
with  the  locus  F^. 

Suppose  a  number  of  such  bodies  to  be  scattered  throughout  the 
medium.  Let  at  first  the  regions  ¥„,  and  F^  be  entirely  outside  the 
space  where  the  bodies  are  situated:  and,  in  making  this  supposition 
we  may,  if  we  please,  suppose  that  the  loci  which  we  are  calhng 
Fa  and  F^,  are  meanwhile  situated  somewhat  farther  from  the  axis 
than  in  our  figure,  that  (for  instance)  F^  is  situated  where  we  have 
drawn  F^,  and  that  F^  is  still  farther  out.  The  bodies  then  tend 
towards  the  poles;  but  the  tendency  may  be  very  small  if,  in 
Fig.  98,  the  curve  and  its  intersecting  straight  hne  do  not  diverge 
very  far  from  one  another  beyond  Fa\    in  other  words,  if,  when 


Fig.  99. 

situated  in  this  region,  the  permeability  of  the  bodies  is  not  very 
much  in  excess  of  that  of  the  medium. 

Let  the  poles  now  tend  to  separate  farther  and  farther  from  one 
another,  the  strength  of  each  pole  remaining  unaltered;  in  other 
words,  let  the  centrosome-foci  recede  from  one  another,-  as  they 
actually  do,  drawing  out  the  spindle-threads  between  them.  The 
loci  Fa,  F^  will  close  in  to  nearer  relative  distances  from  the  poles. 
In  doing  so,  when  the  locus  F^  crosses  one  of  the  bodies,  the  body 
may  be  torn  asunder;  if  the  body  be  of  elongated  shape,  and  be 
crossed  at  more  points  than  one,  the  forces  at  work  will  tend  to 
exaggerate  its  foldings,  and  the  tendency  to  rupture  is  greatest 
when  Fa  is  in  some  median  position  (Fig.  99). 

When  the  locus  Fa  has  passed  entirely  over  the  body,  the  body 
tends  to  move  towards  regions  of  weaker  force;  but  when,  in  turn, 
the  locus  Fi,  has  crossed  it,  then  the  body  again  moves  towards 
regions  of  stronger  force,  that  is  to  say,  towards  the  nearest  pole. 


IV]  AND  STRUCTURE  OF  THE  CELL  321 

And,  in  thus  moving  towards  the  pole,  it  will  do  so,  as  appears 
actually  to  be  the  case  in  the  dividing  cell,  along  the  course  of  the 
outer  hues  of  force,  the  so-called  "mantle-fibres"  of  the  histologist*. 

Such  considerations  as  these  give  general  results,  easily  open  to 
modification  in  detail  by  a  change  of  any  of  the  arbitrary  postulates 
which  have  been  made  for  the  sake  of  simpHcity.  Doubtless  there 
are  other  assumptions  which  would  meet  the  case;  for  instance, 
that  during  the  active  phase  of  the  chromatin  molecule  (when  it  de- 
composes and  sets  free  nucleic  acid)  it  carries  a  charge  opposite  to 
that  which  it  bears  during  its  resting,  or  alkahne  phase ;  and  that  it 
would  accordingly  move  towards  different  poles  under  the  influetice 
of  a  current,  wandering  with  its  negative  charge  in  an  alkahne  fluid 
during  its  acid  phase  to  the  anode,  and  to  the  kathode  during  its 
alkahne  phase.  A  whole  field  of  speculation  is  opened  up  when  we 
begin  to  consider  the  cell  not  merely  as  a  polarised  electrical  field, 
but  also  as  an  electrolytic  field,  full  of  wandering  ions.  Indeed  it 
is  high  time  we  reminded  ourselves  that  we  have  perhaps  been 
deahng  too  much  with  ordinary  physical  analogies:  and  that  our 
whole  field  of  force  within  the  cell  is  of  an  order  of  magnitude  where 
these  grosser  analogies  may  fail  to  serve  us,  and  might  even  play 
us  false,  or  lead  us  astray.  But  our  sole  object  meanwhile,  as  I 
have  said  more  than  once,  is  to  demonstrate,  by  such  illustrations 
as  these,  that,  whatever  be  the  actual  and  as  yet  unknown  modus 
operandi,  there  are  physical  conditions  and  distributions  of  force 
which  could  produce  just  such  phenomena  of  movement  as  we  see 
taking  place  within  the  living  cell.  This,  and  no  more,  is  precisely 
what  Descartes  is  said  to  have  claimed  for  his  description  of  the 
human  body  as  a  "  mechanism  f." 

While  it  can  scarcely  be  too  often  repeated  that  our  enquiry  is 
not  directed  towards  the  solution  of  physiological  problems,  save 
only  in  so  far  as  they  are  inseparable  from  the  problems  presented 
by  the  visible  configurations  of  form  and  structure,  and  while  we 
try,  as  far  as  possible,  to  evade,  the  difficult  question  of  what 

*  •  We  have  not  taken  account  in  the  above  paragraphs  of  the  obvious  fact  that 
the  supposed  symmetrical  field  of  force  is  distorted  by  the  presence  in  it  of  the 
more  or  less  permeable  bodies;  nor  is  it  necessary  for  us  to  do  so,  for  to  that 
distorted  field  the  above  argument  continues  to  apply,  word  for  word. 

t  Michael  Foster,  Lectures  on  the  History  of  Physiology,  1901,  p.  62. 


322  ON  THE  INTERNAL  FORM  [ch. 

particular  forces  are  at  work  when  the  mere  visible  forms  produced 
are  such  as  to  leave  this  an  6pen  question,  yet  in  this  particular 
case  we  have  been  drawn  into  the  use  of  electrical  analogies,  and 
we  are  bound  to  justify,  if  possible,  our  resort  to  this  particular 
mode  of  physical  action.  There  is  an  important  paper  by  R.  S.  LiUie, 
on  the  "Electrical  convection  of  certain  free  cells  and  nuclei*," 
which,  while  I  cannot  quote  it  in  direct  support  of  the  suggestions 
which  I  have  made,  yet  gives  just  the  evidence  we  need  in  order 
to  shew  that  electrical  forces  act  upon  the  constituents  of  the  cell, 
and  that  their  action  discriminates  between  the  two  species  of 
colloids  represented  by  the  cytoplasm  and  the  nuclear  chromatin. 
And  the  difference  is  such  that,  in  the  presence  of  an  electrical 
current,  the  cell  substance  and  the  nuclei  (including  sperm-cells) 
tend  to  migrate,  the  former  on  the  whole  with  the  positive,  the 
latter  with  the  negative  stream :  a  difference  of  electrical  potential 
being  thus  indicated  between  the  particle  and  the  surrounding 
medium,  just  as  in  the  case  of  minute  suspended  particles  of  various 
kinds  in  various  feebly  conducting  media  j*.  And  the  electrical 
difference  is  doubtless  greatest,  in  the  case  of  the  cell  constituents, 
just  at  the  period  of  mitosis:  when  the  chromatin  is  invariably 
in  its  most  deeply  staining,  most  strongly  acid,  and  therefore, 
presumably,  in  its  most  electrically  negative  phase.  In  short,  LilHe 
comes  easily  to  the  conclusion  that  "electrical  theories  of  mitosis 
are  entitled  to  more  careful  consideration  than  they  have  hitherto 
received." 

*  Amer.  J.  Physiol,  viii,  pp.  273-283,  1903  {vide  supra,  p.  314);  cf.  ibid,  xv, 
pp.  46-84,  1905;  xxii,  p.  106,  1910;  xxvii,  p.  289,  1911;  Journ.  Exp.  Zool.  xv, 
p.  23,  1913;   etc. 

t  In  like  manner  Hardy  shewed  that  colloid  particles  migrate  with  the  negative 
stream  if  the  reaction  of  the  surrounding  fluid  be  alkahne,  and  vice  versa.  The 
whole  subject  is  much  wider  than  these  brief  allusions  suggest,  and  is  essentially 
part  of  Quincke's  theory  of  Electrical  Diffusion  or  Endosmosis:  according  to 
which  the  particles  and  the  fluid  in  which  they  float  (or  the  fluid  and  the  capillary 
wall  through  which  it  flows)  each  carry  a  charge:  there  being  a  discontinuity  of 
potential  at  the  surface  of  contact  and  hence  a  field  of  force  leading  to  powerful 
tangential  or  shearing  stresses,  communicating  to  the  particles  a  velocity  which 
varies  with  the  density  per  unit  area  of  the  surface  charge.  See  W.  B.  Hardy's 
paper  on  Coagulation  by  electricity,  Journ.  Physiol,  xxrv,  pp.  288-304,  1899; 
also  Hardy  and  H.  W.  Harvey,  Surface  electric  charges  of  living  cells,  Proc.  E.S. 
(B),  Lxxxiv,  pp.  217-226,  191 1,  and  papers  quoted  therein.  Cf.  also  E.  N.  Harvey's 
observations  on  the  convection  of  unicellular  organisms  in  an  electric  field  (Studies 
on  the  permeability  of  cells,  Journ.  Exp.  Zool.  x,  pp.  508-556,  1911). 


IV] 


AND  STRUCTURE  OF  THE  CELL 


323 


Among  other  investigations  all  leading  towards  the  same  general 
conclusion,  namely  that  differences  of  electric  potential  play  their 
part  in  the  phenomena  of  cell  division,  I  would  mention  a  note- 
worthy paper  by  Ida  H.  Hyde*,  in  which  the  writer  shews  (among 
other  important  observations)  that  not  only  is  there  a  measurable 
difference  of  potential  between  the  animal  and  vegetative  poles  of 
a  fertilised  egg  (Fundulus,  toad,  turtle,  etc.),  but  also  that  this 
difference  fluctuates,  or  actually  reverses  its  direction,  periodically, 
at  epochs  coinciding  with  successive  acts  of  segmentation  or  other 


Fig.  100.  Final  stage  in  the  first  seg- 
mentation of  the  egg  of  Cerebra- 
tidus.  From  Prenant,  after  Coe*. 


Fig.  101.     Diagram  of  field  of  force 
with  two  similar  poles. 


important  phases  in  the  development  of  the'  eggt;  just  as  other 
physical  rhythms,  for  instance,  in  the  production  of  COg ,  had  already 
been  shewn  to  do.  Hence  we  need  not  be  surprised  to  find  that  th^ 
"materialised"  hues  of  force,  which  in  the  earlier  stages  form  the 


*  On  differences  in  electrical  potential  in  developing  eggs,  Amer.  Journ.  Physiol. 
XII,  pp.  241-275,  1905.  This  paper  contains  an  excellent  summary,  for  the  time 
being,  of  physical  theories  of  the  segmentation  of  the  cell. 

t  Gray  has  demonstrated  a  temporary  increase  of  electrical  conductivity  in 
sea-urchin  eggs  during  the  process  of  fertilisation,  and  ascribes  the  changes  in 
resistance  to  polarisation  of  the  surface:  Electrical  conductivity  of  echinoderm 
eggs,  etc.,  Phil.  Trans.  (B),  ccvii,  pp.  481-529,  1916. 


324 


ON  THE  INTERNAL  FORM 


[CH. 


convergent  curves  of  the  spindle,  are  replaced  in  the  later  phases  of 
caryokinesis  by  divergent  curves,  indicating  that  the  two  foci,  which 
are  marked  out  in  the  field  by  the  divided  and  reconstituted  nuclei, 
are  now  ahke  in  their  polarity*  (Figs.  100,  101). 

The  foregoing  account  is  based  on  the  provisional  assumption 
that  the  phenomena  of  caryokinesis  are  analogous  to  those  of  a 
bipolar  electrical  field — a  comparison  which  seems  to  offer  a  helpful 
and  instructive  series  of  analogies.  But  there  are  other  forces  which 
lead  to  similar  configurations.  For  instance,  some  of  Leduc's 
diffusion-experiments  offer  very  remarkable  analogies  to  the  dia- 
grammatic  phenomena   of   caryokinesis,  as   shewn   in   Fig.   102  f. 


Fig.  102.     Artificial  caryokinesis  (after  Leduc),  for  comparison  with  Fig.  88,  p.  299. 

Here  we  have  two  identical  (not  opposite)  poles  of  osmotic  con- 
centration, formed  by  placing  a  drop  of  Indian  ink  in  salt  water, 
and  then  on  either  side  of  this  central  drop,  a  hypertonic  drop  of 
salt  solution  more  lightly  coloured.  On  either  side  the  pigment  of 
the  central  drop  has  been  drawn  towards  the  focus  nearest  to  it; 
but  in  the  middle  line,  the  pigment  is  drawn  in  opposite  directions 
by  equal  forces,  and  so  tends  to  remain  undisturbed,  in  the  form  of 
an  "equatorial  plate." 

To  account  for  the  same  mitotic  phenomena  an  elegant  hypothesis 
has  been  put  forward  by  A.  B.  Lamb  J,  and  developed  by  Graham 

*  W.  R.  Coe,  Maturation  and  fertilisation  of  the  egg  of  Cerebratulus,  Zool. 
Jahrbiicher  {Anat.  Abth.),  xii,  pp,  425^76,  1899. 

t  Op.  cit.  pp.  110  and  91. 

t  A.  B.  Lamb,  A  new  explanation  of  the  mechanism  of  mitosis,  Journ.  Exp.  Zool. 
V,  pp.  27-33.  1908. 


IV]  AND  STRUCTURE  OF  THE  CELL  325 

Cannon*.  It  depends  on  certain  investigations  of  the  Bjerknes, 
father  and  sonf,  which  prove  that  bodies  pulsating  or  oscillating  J 
in  a  fluid  set  up  a  field  of  force  precisely  comparable  with  the  lines 
of  force  in  a  magnetic  field.  Certain  old  and  even  familiar  observa- 
tions had  pointed  towards  this  phenomenon.  Guyot  had  noticed 
that  bits  of  paper  were  attracted  towards  a  vibrating  tuning-fork; 
and  Schellbach  found  that  a  sounding-board  so  acts  on  bodies  in  its 
neighbourhood  as  to  attract  those  which  are  heavier  and  repel  those 
which  are  lighter  than  the  surrounding  medium;  in  air  bits  of 
paper  are  attracted  and  a  gas-flame  is  repelled.  To  explain  these 
simple  observations,  Bjerknes  experimented  with  Httle  drums 
attached  to  an  automatic  bellows.  He  found  that  two  bodies  in 
a  fluid  field,  synchronously  pulsating  or  synchronously  oscillating, 
repel  one  another  when  their  oscillations  are  in  the  same  phase,  or 
their  pulsations  are  in  opposite  phase;  and  vice  versa:  while  other 
particles,  floating  passively  in  the  same  fluid,  tend  (as  Schellbach 
had  observed  before)  to  be  attracted  or  repulsed  according  as  they 
are  heavier  or  lighter  than  the  fluid  medium.  The  two  bodies 
behave  towards  one  another  like  two  electrified  bodies,  or  like  two 
poles  of  a  magnet;  we  are  entitled  to  speak  of  them  as  "hydro- 
dynamic  poles,"  we  might  even  call  them  " hydrodynamic  magnets" ; 
and  pursuing  the  analogy,  we  may  call  the  heavy  bodies  para- 
magnetic, and  the  light  ones  diamagnetic  with  regard  to  them. 
Lamb's  hypothesis  then,  and  Cannon's,  is  that  the  centrosomes  act 
as  "hydrodynamic  magnets."  The  explanation  depends  on  oscilla- 
tions which  have  never  been  seen,  in  centrosomes  which  are  not 
always  to  be  discovered.  But  it  brings  together  certain  curious 
analogies,  and  these,  where  we  know  so  little,  may  be  worth 
reflecting  on. 

If  we  assume  that  each  centrosome  is  endowed  with  a  vibratory 
motion  as  it  floats  in  the  semi-fluid  colloids,  or  hydrosols  (to  use 
Graham's  word)  of  the  cell,  we  ma;^  take  it  that  the  visible  intra- 
cellular  phenomena   will  be   much   the   same  as   those   we   have 

*  Op.  cit.  Cf.  also  Gertrud  Woken,  Zur  Physik  der  Kernteilung,  Z.  f.  allg,  Physiol. 
XVIII,  pp.  39-57,  1918. 

t  V.  Bjerknes,  Vorlesungen  liber  hydrodynamische  Fernkrdjte,  nach  C.  A.  Bjerknes^ 
Theorie,  Leipzig,  1900. 

J  A  body  is  said  to  pulsate  when  it  undergoes  a  rhythmic  change  of  volume; 
it  oscillates  when  it  undergoes  a  rhythmic  change  of  place. 


326  ON  THE  INTERNAL  FORM  [ch. 

described  under  an  electrical  hypothesis;  the  lines  of  force  will 
have  the  same  distribution,  and  such  movements  as  the  chromo- 
somes undergo,  and  such  symmetrical  configurations  as  they  assume, 
may  be  accounted  for  under  the  one  hypothesis  pretty  much  as 
under  the  other.  There  are  however  other  phenomena  accompanying 
mitosis,  such  as  Chambers's  astral  currents  and  certain  local  changes 
in  the  viscosity  of  the  egg,  which  are  more  easily  explained  by  the 
hydrodynamic  theory. 

We  may  assume  that  the  cytoplasm,  however  complex  it  may  be, 
is  but  a  sort  of  microscopically  homogeneous  emulsion  of  high 
dispersion,  that  is  to  say  one  in  which  the  minute  particles  of  one 
phase  are  widely  scattered  throughout,  and  freely  mobile  in,  the 
other;  and  this  indeed  is  what  is  meant  by  caUing  it  a  hydrosol. 
Let  us  assume  also  that  the  particles  are  a  little  less  dense  than  the 
continuous  phase  in  which  they  are  dispersed;  and  assume  lastly 
(it  is  not  the  easiest  of  our  assumptions)  that  these  ultra-minute 
particles  will  be  affected,  just  as  are  the  grosser  ones,  by  the  forces 
of  the  hydrodynamic  field. 

All  this  being  so,  the  disperse  particles  will  be  repelled  from  the 
oscillating  centrosome,  with  a  force  which  falls  off  very  rapidly,  for 
Bjerknes  tells  us  that  it  varies  inversely  as  the  seventh  power  of 
the  distance;  a  round  clear  field,  hke  a  drop  or  a  bubble,  will  be 
formed  round  the  centrosome ;  and  the  disperse  particles,  expelled 
from  this  region,  will  tend  to  accumulate  in  a  crowded  spherica- 
zone  immediately  beyond  it.  Outside  of  this  again  they  will  con- 
tinue to  be  repulsed,  but  slowly,  and  we  may  expect  a  second  and 
lesser  concentration  at  the  periphery  of  the  cell.  A  clear  central 
mass,  or  "centrosphere,"  will  thus  come  into  being;  and  the 
surrounding  cytoplasm  will  be  rendered  denser  and  more  viscous, 
especially  close  around  the  centrosphere  and  again  peripherally,  by 
condensation  of  the  disperse  particles.  Moreover,  all  outward 
movements  of  these  lighter  particles  entail  inward  movements  of 
the  heavier,  which  (by  hypothesis)  are  also  the  more  fluid ;  stream- 
lines or  visible  currents  will  flow  towards  the  centre,  giving  rise  to 
the  star-shaped  "aster,"  and  the  best  accounts  of  the  sea-urchin's 
egg*  tally  well  with  what  is  thus  deduced  from  the  hydrodynamic 

*  Cf.  R.  Chambers,  in  Journ.  Exp.  Zool.  xxni,  p.  483,  1917;  Trans.  R.S.  Canada, 
XII,  1918;   Journ.  Gen.  Physiol,  ii,  1919. 


IV]  AND  STRUCTURE  OF  THE  CELL  327 

hypothesis.  The  round  drop  of  clear  fluid  which  forms  the  centre 
of  the  aster  grows  as  the  aster  grows,  fluid  streaming  towards  it 
from  all  parts  of  the  cell  along  the  channels  of  the  astral  rays. 
The  cytoplasni  between  the  rays  is  in  the  gel  state,  but  gradually 
passes  into  a  sol  beyond  the  confines  of  the  aster.  Seifritz  asserts 
that  the  substance  of  the  centrosphere  is  *'not  much  more  viscous 
than  water,"  but  that  the  wedges  of  cytoplasm  between  the  inwardly 
directed  streams  are  stiff  and  viscous*. 

After  the  centrosome  divides  we  have  two  oscillating  bodies 
instead  of  one;  they  tend  to  repel  one  another,  and  pass  easily 
through  the  fluid  centrosphere  to  the  denser  layer  around.  But 
now  the  new  centrosomes,  on  opposite  sides  of  the  centrosphere, 
repel,  each  on  its  own  side,  the  disperse  particles  of  the  denser  zone ; 
and  two  new  asters  are  formed,  their  rays  marked  by  the  streams 
coursing  inwards  to  the  centrosome-foci.  Thus  the  amphiaster 
comes  into  being;  it  is  not  that  the  old  aster  divides,  as  a  definite 
entity;  but  the  old  aster  ceases  to  exist  when  its  focus  is  disturbed, 
and  about  the  new  foci  new  asters  are  necessarily  and  automatically 
developed.  Again  this  hypothetic  account  taUies  well  with  Chambers's 
description. 

The  same  attractions  and  repulsions  should  be  manifested,  perhaps 
better  still,  in  whatsoever  bodies  he  or  float  within  the  cell,  whether 
liquid  or  solid,  oil-globules,  yolk-particlea,  mitochondria,  chromo- 
somes or  what  not.  A  zoned,  concentric  arrangement  of  yolk- 
globules  is  often  seen  in  the  egg,  with  the  centrosome  as  focus; 
and  in  certain  sea-urchin  eggs  the  mitochondria  gather  around 
the  centrosome  while  the  amphiaster  is  forming,  collecting  together 
in  that  very  zone  to  which  Chambers  ascribes  a  semi-rigid  or  viscous 
consistency!.  The  Golgi  bodies  found  in  various  germ-cells  are  at 
first  black  rod-hke  bodies  embedded  in  the  centrosphere ;  they 
undergo  changes  and  complex  movements,  now  scattering  through 
the  cytoplasm  and  anon  crowding  again  around  the  centrosome. 
Some  periodic  change  in  the  density  of  these  bodies  compared  with 

♦  Cf.  W.  Seifritz,  Some  physical  properties  of  protoplasm,  Ann.  Bot.  xxxv, 
1921.  Wo.  Ostwald  and  M.  H.  Fischer  had  thought  that  the  astral  rays  were 
due  to  local  changes  of  the  plasma-sol  into  a  gel,  Zur  physikal.  chem.  Theorie  der 
Befruchtung,  Pfluger's  Archiv,  cvi,  pp.  2^3-266,  1905. 

t  Cf.  F.  Vejdovsky  and  A.  Mrazek,  Umbildung  des  Cytoplasma  wahrend  der 
Befruchtung  und  Zelltheilung,  Arch.  f.  mikr.  Anat.  LXii.  431-579,  1903. 


328  ON  THE  INTERNAL  FORM    .  [ch. 

that  of  the  medium  in  which  they  He  seems  all  that  is  required 
to  account  for  their  excursions;  and  such  changes  of  density  are 
not  only  of  likely  occurrence  during  the  active  chemical  operations 
associated  with  fertilisation  and  division,  but  are  in  all  probability 
inseparable  from  the  changes  in  viscosity  which  are  known  to 
occur*.  The  movements  and  arrangements  of  the  chromosomes, 
already  described,  may  be  easily  accounted  for  if  we  postulate,  in 
addition  to  their  repulsion  from  the  oscillating  centrosomes,  induced 
oscillations  in  themselves  such  as  to  cause  them  to  attract  one  another. 

The  well-defined  length  of  the  spindle  and  the  position  of  equili- 
brium in  which  it  comes  to  rest  may  be  conceived  as  resultants  of 
the  several  mutual  repulsions  of  the  centrosomes  by  one  another, 
by  the  chromosomes  or  other  lighter  material  of  the  equatorial  plate, 
and  again  by  such  lighter  material  as  may  have  accumulated  at  the 
periphery  of  the  egg ;  the  first  two  of  these  will  tend  to  lengthen  the 
spindle,  the  last  to  shorten  it;  and  the  last  will  especially  affect  its 
position  and  direction.  When  Chambers  amputated  part  of  an 
amphiastral  egg,  the  remains  of  the  amphiaster  disappeared,  and 
then  came  into  being  again  in  a  new  and  more  symmetrical  position ; 
it  or  its  centrosomal  focus  had  been  symmetrically  repelled,  we  may 
suppose,  by  the  fresh  surface.  Hertwig's  law  that  the  spindle-axis 
tends  to  lie  in  the  direction  of  the  largest  mass  of  protoplasm,  in 
other  words  to  point  where  the  cell-surface  lies  farthest  off  and  its 
repulsion  is  least  felt,  may  likewise  find  its  easy  explanation. 

Between  these  hypotheses  we  may  choose  one  or  other  (if  we 
choose  at  all),  according  to  our  judgment.  As  Henri  Poincare  tells 
us,  we  never  know  that  any  one  physical  hypothesis  is  true,  we  take 
the  simplest  we  can  find;  and  this  we  call  the  guiding  principle  of 
simphcity !  In  this  case,  the  hydrodynamic  hypothesis  is  a  simple 
one;  but  it  all  rests  on  a  hypothetic  oscillation  of  the  centrosomes, 
which  has  never  been  witnessed.  Bayliss  has  shewn  that  precisely 
such  reversible  states  of  gelation  as  we  have  been  speaking  of  as 

*  Cf.  G.  Odquist,  Viscositatsanderungen  des  Zellplasmas  wahrend  der  ersten 
Entwicklungsstufen  des  Froscheies,  Arch.  f.  Entw.  Mech.  ia,  pp.  610-624,  1922; 
A.  Gurwitsch,  Pramissen  und  anstossgebende  Faktoren  der  Furchung  und 
Zelltheilung,  Arch.  f.  Zellforsch.  n,  pp.  495-548,  1909;  L.  V.  Heilbrunn, 
Protoplasmic  viscosity-changes  during  mitosis,  Journ.  Exp.  Zool.  xxxiv,  pp.  417-447, 
1921 ;  ibid,  xliv,  pp.  255-278,  1926;  E.  Leblond,  Passage  de  I'etat  de  gel  a  I'etat  de  sol 
dans  le  protoplasme  vivant,  C.R.  Soc.  Biol,  lxxxii,  p.  1150;   of.  ibid.  p.  1220;  etc. 


rv]  AND  STRUCTURE  OF  THE  CELL  329 

"periodic  changes  in  viscosity"  may  be  induced  in  living  protoplasm 
by  electrical  stimulation*.  On  the  other  hand,  the  fact  that  the 
hydrodynamic  forces  fall  off  as  fast  as  they  do  with  increasing 
distance  Hmits  their  efficacy ;  and  the  minute  disperse  particles 
must,  under  Stokes's  law,  be  slow  to  move.  Lastly,  it  may  well  be 
(as  Lillie  has  urged)  that  such  work  as  his  own,  or  Ida  Hyde's,  or 
Gray's,  on  change  of  potential  in  developing  eggs,  taken  together 
with  that  of  many  others  on  the  behaviour  of  colloid  particles  in  an 
electrical  field,  has  not  yet  been  followed  out  in  all  its  consequences, 
either  on  the  physical  or  the  physiological  side  of  the  problem. 

But  to  return  to  our  general  discussion. 

As  regards  the  actual  mechanical  division  of  the  cell  into  two 
halves,  we  shall  see  presently  that,  in  certain  cases,  such  as  that 
of  a  long  cylindrical  filament,  surface-tension,  and  what  is  known 
as  the  principle  of  "minimal  areas,"  go  a  long  way  to  explain  the 
mechanical  process  of  division;  and  in  all  cells  whatsoever,  the 
process  of  division  must  somehow  be  explained  as  the  result  of  a 
conflict  between  surface-tension  and  its  opposing  forces.  But  in 
such  a  case  as  our  spherical  cell,  it  is  none  too  easy  to  see  what 
physical  cause  is  at  work  to  disturb  its  equiUbrium  and  its  integrity. 

The  fact  that  when  actual  division  of  the  cell  takes  place,  it  does 
so  at  right  angles  to  the  polar  axis  and  precisely  in  the  direction 
of  the  equatorial  plane,  would  lead  us  to  suspect  that  the  new 
surface  formed  in  the  equatorial  plane  sets  up  an  annular  tension, 
directed  inwards,  where  it  meets  the  outer  surface  layer  of  the  cell 
itself.  But  at  this  point  the  problem  becomes  more  comphcated. 
Before  we  can  hope  to  comprehend  it,  we  shall  have  not  only  to 
enquire  into  the  potential  distribution  at  the  surface  of  the  cell  in 
relation  to  that  which  we  have  seen  to  exist  in  its  interior,  but  also 
to  take  account  of  the  differences  of  potential  which  the  material 
arrangements  along  the  lines  of  force  must  themselves  tend  to 
produce.  Only  thus  can  we  approach  a  comprehension  of  the 
balance  of  forces  which  cohesion,  friction,  capillarity  and  electrical 
distribution  combine  to  set  up. 

The  manner  in  which  we  regard  the  phenomenon  would  seem  to 

*  W.  M.  Bayliss,  Reversible  gelation  in  living  protoplasm,  Proc.  R.S.  (B),  xci, 
pp.  196-201,  1920. 


PO  ON  THE  INTERNAL  FORM  [ch. 

turn,  in  great  measure,  upon  whether  or  no  we  are  justified  in 
assuming  that,  in  the  hquid  surface-film  of  a  minute  spherical  cell, 
local  and  symmetrically  localised  differences  of  surface-tension  are 
likely  to  occur.  If  not,  then  changes  in  the  conformation  of  the 
cell  such  as  lead  immediately  to  its  division  must  be  ascribed  not 
to  local  changes  in  its  surface-tension,  but  rather  to  direct  changes 
in  internal  pressure,  or  to  mechanical  forces  due  to  an  induced 
surface-distribution  of  electrical  potential.  We  have  little  reason  to 
be  sceptical ;  in  fact  we  now  know  that  the  cell  is  so  far  from  being 
chemically  and  physically  homogeneous  that  local  variations  in  its 
surface-tension  are  more  than  likely,  they  are  certain  to  occur. 

Biitschh  suggested  more  than  sixty  years  ago  that  cell-division 
was  brought  about  by  an  increase  of  surface-tension  in  the  equatorial 
region  of  the  cell ;  and  the  suggestion  was  the  more  remarkable  that 
it  was  (I  beheve)  the  very  first  attempt  to  invoke  surface-tension 
as  a  factor  in  the  physical  causation  of  a  biological  phenomenon*. 
An  increase  of  equatorial  tension  would  cause  the  surface-area  there 
to  diminish,  and  the  equator  to  be  pinched  in,  but  the  total  surface- 
area  of  the  cell  would  be  increased  thereby,  and  the  two  effects 
would  strike  a  balance  f.  But,  as  Biitschh  knew  very  well,  the 
surface-tension  change  would  not  stand  alone;  it  would  bring  other 
phenomena  in  its  train,  currents  would  tend  to  be  set  up,  and 
tangential  strains  would  be  imposed  on  the  cell-membrane  or  cell- 
surface  as  a  whole.  The  secondary  if  not  the  direct  effects  of 
increased  equatorial  tension  might,  after  all,  suffice  for  the  division  of 
the  cell.  It  was  Loeb,  in  1895,  who  first  shewed  that  streaming  went 
on  from  the  equator  towards  the  divided  nuclei.  To  the  violence 
of  these  streaming  movements  he  attributed  the  phenomenon  of 
division,  and  many  other  physiologists  have  adopted  this  hypo- 
thesis J.     The  currents  of  which  Loeb  spoke  call  for  counter-currents 

*  0.  Butschli,  tJber  die  ersten  Entwicklungsvorgange  der  Eizelle,  Abh. 
Senckenberg.  naturf.  Gesellsch.  x,  1876;  Uber  Plasmastrqmungen  bei  der  Zell- 
theilung,  Arch.  f.  Entw.  Mech.  x,  p.  52,  1900.  Ryder  ascribed  the  earyokinetic 
figures  to  surface-tension  in  his  Dynamics  in  Evolution,  1894. 

t  A  relative,  not  positive,  increase  of  surface-tension,  was  part  of  Giardina's 
hypothesis:   Note  sul  mecanismo  della  divisione  cellulare,  Anat.  Anz.  xxi.  1902. 

J  J.  Loeb,  Amer.  Journ.  Physiol,  vi,  p.  432,  1902;  E.  G.  Conklin,  Protoplasmic 
movements  as  a  factor  in  differentiation,  Wood's  Hole  Biol.  Lectures,  p.  69,  etc., 
1898-99;  J.  Spek,  Oberflachenspannungsdifferenzen  als  eine  Ursache  der  Zell- 
teilung,  Arch.f.  Entw.  Mech.  xliv,  pp.  54-73,  1918. 


IV]  AND  STRUCTURE  OF  THE  CELL  331 

towards  the  equator,  in  or  near  the  surface  of  the  cell;  and  theory 
and  observation  both  indicate  that  precisely  such  currents  are  bound 
to  be  set  up  by  the  surface-energy  involved  in  the  increase  of 
equatorial  tension. 

An  opposite  view  has  been  held  by  some,  and  especially  by 
T.  B.  Robertson*.  Quincke  had  shewn  that  the  formation  of  soap 
at  the  surface  of  an  oil-droplet  lowers  the  surface-tension  of  the 
latter,  and  that  if  the  saponification  be  local,  that  part  of  the  surface 
tends  to  enlarge  and  spread  out  accordingly.  Robertson,  in  a  very 
curious  experiment,  found  that  by  laying  a  thread,  moistened  with 
dilute  caustic  alkali  or  merely  smeared  with  soap,  across  a  drop  of 
olive  oil  afloat  in  water,  the  drop  at  once  divided  into  two.  A 
vast  amount  of  controversy  has  arisen  over  this  experiment,  but 
Spek  seems  to  have  shewn  conclusively  that  it  is  an  exceptional 
case. 

In  a  drop  of  olive-oil,  balanced  in  water f  and  touched  anywhere 
with  an  alkaH,  there  is  so  copious  a  formation  of  lighter  soaps  that 
di^erences  of  density  tend  to  drag  the  drop  in  two.  But  in  the 
case  of  other  oils  (and  especially  the  thinner  oils,  such  as  oil  of 
bergamot)  the  saponified  portion  bulges,  as  theory  directs;  and 
when  the  alkali  is  applied  to  two  opposite  poles  the  equatorial 
region  is  pinched  in,  as  McClendonJ,  in  opposition  to  Robertson, 
had  found  it  to  do.  Conversely,  if  an  alkaline  thread  be  looped 
around  the  drop,  the  zone  of  contact  bulges,  and  instead  of  dividing 
at  the  equator  the  drop  assumes  a  lens-like  form. 

We  may  take  it  then  as  proven  that  a  relative  increase  of  equatorial 
surface-tension,  whether  in  oil-drops,  mercury-globules  or  living 
cells,  does  lead,  or  tend  to  lead,  to  an  equatorial  constriction.  In 
all  cases  a  system  of  surface-currents  is  set  up  among  the  fluid  drops 
towards  the  zone  of  increased  tension ;  and  an  axial  counter-current 
flows  towards  the  pole  or  poles  of  lowered  tension.  Precisely  such 
currents  have  been  observed  to  run  in  various  eggs  (especially  of 

♦  T.  B.  Robertson,  Note  on  the  chemical  mechanics  of  cell-division,  Arch.  f. 
Entw.  Mech.  xxvn,  p.  29,  1909;  xxxii,  p.  308,  1911;  xxxv,  p.  402,  1913.  Cf. 
R.  S.  Lillie,  Joum.-Exp.  Zool.  xxi,  pp.  369^02,  1916;   McClendon,  loc.  cit.;   etc. 

t  In  these  experiments,  and  in  many  of  Quincke's,  a  little  chloroform  is  added 
to  the  oil,  in  order  to  bring  its  density  as  near  as  may  be  to  that  of  water. 

X  J.  F.  McClendon,  Note  on  the  mechanics  of  cell -division,  Arch.  f.  Entw.  Mech. 
xxxrv,  pp.  263-266,  1912. 


332  ON  THE  INTERNAL  FORM  [ch. 

certain  Nematodes)  during  division  of  the  cell;  but  if  the  })rocess 
be  slow,  more  than  7  or  8  minutes  long,  the  slow  currents  become 
hard  to  see.  Various  contents  of  the  cell  are  transported  by  these 
currents,  and  clear,  yolk-free  polar  caps  and  equatorial  accumula- 
tions of  yolk  and  pigment  are  among  the  various  manifestations  of 
the  phenomenon.  The  extrusion  of  a  polar  body,  at  a  small  and 
sharply  defined  region  of  lowered  tension,  is  a  particular  case  of  the 
same  principle*. 

But  purely  chemical  changes  are  not  of  necessity  the  fundamental 
cause  of  alteration  in  the  surface-tension  of  the  egg,  for  the  action 
of  electrolytes  on  surface-tension  is  now  well  known  and  easily 
demonstrated.  So,  according  to  other  views  than  those  with  which 
we  have  been  dealing,  electrical  charges  are  sufficient  in  themselves 
to  account  for  alterations  of  surface-tension,  and  in  turn  for  that 
protoplasmic  streaming  which,  as  so  many  investigators  agree, 
initiates  the  segmentation  of  the  eggf.  A  great  part  of  our  difficulty 
arises  from  the  fact  that  in  such  a  case  as  this  the  various  pheno- 
mena are  so  entangled  and  apparently  concurrent  that  it  is  hard 
to  say  which  initiates  another,  and  to  which  this  or  that  secondary 
phenomenon  may  be  considered  due.  Of  recent  years  the  pheno- 
menon of  adsorption  has  been  adduced  (as  we  have  already  briefly 
said)  in  order  to  account  for  many  of  the  events  and  appearances 
which  are  associated  with  the  asymmetry,  and  lead  towards  the 
division,  of  the  cell.  But  our  short  discussion  of  this  phenomenon 
may  be  reserved  for  another  chapter. 

However,  we  are  not  directly  concerned  here  with  the  phenomena 
of  segmentation  or  cell-division  in  themselves,  except  only  in  so  far 
as  visible  changes  of  form  are  capable  of  easy  and  obvious  correla- 
tion with  the  play  of  force.  The  very  fact  of  "development" 
indicates  that,  while  it  lasts,  the  equihbrium  of  the  egg  is  never 
complete  J.  And  the  gist  of  the  matter  is  that,  if  you  have  caryo- 
kinetic  figures  developing  inside  the  cell,  that  of  itself  indicates  that 
the  dynamic  system  and  the  localised  forces  arising  from  it  are  in 

*  J.  Spek,  loc.  cit.  pp.  108-109. 

t  Cf.  D'Arsonval,  Relation  entre  la  tension  superficielle  et  certains  phenomenes 
electriques  d'origine  animale.  Arch,  de  Physiol,  i,  pp.  460-472,  1889;  Ida  H.  Hyde, 
op.  cit.  p.  242. 

I  Cf.  Plateau's  remarks  (Statique  des  liquides,  ii,  p.  154)  on  the  tendency  towards 
equilibrium,  rather  than  actual  equilibrium,  in  many  of  his  systems  of  soap-films. 


IV]  AND  STRUCTURE  OF  THE  CELL  333 

gradual  alteration;  and  changes  in   the  outward  configuration  of 
the  system  are  bound,  consequently,  to  take  place. 

Perhaps  we  may  simplify  the  case  still  more.  We  have  learned 
many  things  about  cell-division,  but  we  do  not  know  much  in  the 
end.  We  have  dealt,  perhaps,  with  too  many  related  phenomena, 
and  failed  because  we  tried  to  combine  and  account  for  them  all. 
A  physical  problem,  still  more  a  mathematical  one,  wants  reducing 
to  its  simplest  terms,  and  Dr  Rashevsky  has  simplified  and  general- 
ised the  problem  of  cell-division  (or  division  of  a  drop)  in  a  series  of 
papers,  which  still  outrun  by  far  the  elementary  mathematics  of 
this  book.  If  we  cannot  follow^  him  in  all  he  does,  we  may  find 
useful  lessons  in  his  way  of  doing  it.  Cells  are  of  many  kinds;  they 
differ  in  size  and  shape,  in  visible  structure  and  chemical  com- 
position. Most  have  a  nucleus,  some  few  have  none;  most  need 
oxygen,  some  few  do  not;  some  metabolise  in  one  way,  some  in 
another.  What  small  residuum  of  properties  remains  common  to 
them  all?  A  living  cell  is  a  little  fluid  (or  semi-fluid)  system,  in 
which  work  is  being  done,  physical  forces  are  in  operation  and 
chemical  changes  are  going  on.  It  is  in  such  intimate  relation  with 
the  world  outside — its  own  milieu  interne  with  the  great  ynilieu 
externe — that  substances  are  continually  entering  the  cell,  some  to 
remain  there  and  contribute  to  its  growth,  some  to  pass  out  again 
with  loss  of  energy  and  metabolic  change.  The  picture  seems 
simplicity  itself,  but  it  is  less  simple  than  it  looks.  For  on  either 
side  of  the  boundary- wall,  both  in  the  adjacent  medium  and  in  the 
living  protoplasm  within,  there  will  be  no  uniformity,  but  only 
degrees  of  activity,  and  gradients  of  concentration.  Substances 
which  are  being  absorbed  and  consumed  will  diminish  from  periphery 
to  centre;  those  which  are  diffusing  outwards  have  their  greatest 
concentration  near  the  centre,  decrease  towards  the  periphery,  and 
diminish  further  with  increasing  distance  in  the  near  neighbourhood  of 
the  system.  Size,  shape,  diffusibility,  permeability,  chemical  properties 
of  this  and  that,  may  affect  the  gradients,  but  in  the  living  cell  the 
interchanges  are  always  going  on,  and  the  gradients  are  always  there  *. 

*  Outward  diffusion  makes  one  of  the  many  contrasts  between  cell-growth  and 
crystal-growth.  But  the  diffusion^gradients  round  a  growing  crystal  are  far  more 
complicated  than  was  once  supposed.  Cf.  W.  F.  Berg,  Crystal  growth  from 
solutions,  Proc.  R.8.  (A),  clxiv,  pp.  79-95,  1938. 


334  ON  THE  INTERNAL  FORM  [ch. 

If  the  cell  be  homogeneous,  taking  in  and  giving  out  at  a  constant 
rate  in  a  uniform  way,  its  shape  will  be  spherical,  the  concentration- 
field  of  force,  or  concentration-field,  will  likewise  have  a  spherical 
symmetry,  and  the  resultant  force  will  be  zero.  But  if  the  symmetry 
be  ever  so  little  disturbed,  and  the  shape  be  ever  so  little  deformed, 
then  there  will  be  forces  at  work  tending  to  increase  the  deformation, 
and  others  tending  to  equalise  the  surface-tension  and  restore  the 
spherical  symmetry,  and  it  can  be  shewn  that  such  agencies  are 
within  the  range  of  the  chemistry  of  the-  cell.  Since  surface- 
tension  becomes  more  and  more  potent  as  the  size  of  the  drop 
diminishes,  it  follows  that  (under  fluid  conditions)  the  smallest 
solitary  cells  are  least  likely  to  depart  from  a  spherical  shape,  and 
that  cell-division  is  only  likely  to  occur  in  cells  above  a  certain 
critical  order  of  magnitude;  and  using  such  physical  constants 
as  are  available,  Rashevsky  finds  that  this  critical  magnitude 
tallies  fairly  well  with  the  average  size  of  a  living  cell.  The  more 
important  lesson  to  learn,  however,  is  this,  that,  merely  by  virtue 
of  its  metabolism,  every  cell  contains  within  itself  factors  which  may 
lead  to  its  division  after  it  reaches  a  certain  critical  size. 

There  are  simple  corollaries  to  this  simple  setting  of  the  case. 
Since  unequal  concentration-gradients  are  the  chief  cause  which 
renders  non-spherical  shapes  of  cell  possible,  and  these  last  only  so 
long  as  the  cell  lives  and  metabolises,  it  follows  that,  as  soon  as  the 
gradients  disappear,  whether  in  death  or  in  a  "resting-stage",  the 
cell  reverts  to  a  spherical  shape  and  symmetry.  Again,  not  only  is 
there  a  critical  size  above  which  cell-division  becomes  possible,, 
and  more  and  more  probable,  but  there  must  also  be  a  size  beyond 
which  the  cell  is  not  likely  to  grow.  For  the  "specific  surface" 
decreases,  the  metabolic  exchanges  diminish,  the  gradients  become 
less  steep,  and  the  rate  of  growth  decreases  too ;  there  must  come 
a  stage  where  anabolism  just  balances  katabolism,  and  growth 
ceases  though  life  goes  on.  When  streaming  currents  are  visible 
within  the  cell,  they  seem  to  complicate  the  problem;  but  after  all, 
they  are  part  of  the  result,  and  proof  of  the  existence,  of  the  gradients 
•we  have  described.  In  any  further  account  of  Rashevsky 's  theories 
the  mathematical  difficulties  very  soon  begin.  But  it  is  well  to 
realise  that  pure  theory  often  carries  the  mathematical  physicist  a 
long  way ;  and  that  higher  and  higher  powers  of  the  microscope,  and 


IV]  AND  STRUCTURE  OF  THE  CELL  335 

greater  and  greater  histological  skill  are  not  the  one  and  only  way 
to  study  the  physical  forces  acting  within  the  cell  *. 

As  regards  the  phenomena  of  fertihsation,  of  the  union  of  the 
spermatozoon  with  the  "pronucleus"  of  the  egg,  we  might  study 
these  also  in  illustration,  up  to  a  certain  point,  of  the  forces  which 
are  more  or  less  manifestly  at  work.  But  we  shall  merely  take,  as 
a  single  illustration,  the  paths  of  the  male  and  female  pronuclei,  as 
they  travel  to  their  ultimate  meeting-place. 

The  spermatozoon,  when  within  a  very  short  distance  of  the  egg- 
cell,  is  attracted  by  it,  the  same  attraction  being  further  manifested 
in  a  small  conical  uprising  of  the  surface  of  the  eggt.  The  nature 
of  the  attractive  force  has  been  much  disputed.  Loeb  found  the 
spermatozoon  to  be  equally  attracted  by  other  substances,  even  by 
a  bead  of  glass.  It  has  been  held  also  that  the  attraction  is 
chemotropic,  some  substance  being  secreted  by  the  egg  which  drew 
the  sperm  towards  it:  just  as  Pfeifer,  having  shewn  that  maUc  acid 
has  an  attraction  for  fern-antheridia,  supposed  this  substance  to 
play  its  attractive  part  within  the  mucus  of  the  archegonia.  Again, 
the  chemical  secretion  may  be  neither  attractive  nor  directive,  but 
yet  play  a  useful  part  in  activating  the  spermatozoa.  However 
that  may  be,  Gray  has  shewn  reason  to  believe  that  an  electromotive 
force  is  developed  in  the  contact  between  active  spermatozoon  and 
inactive  ovum;  and  that  it  is  the  electrical  change  so  set  up,  and 
almost  instantaneously  propagated,  which  precludes  the  entry  of 
another  spermatozoon  J.  Whatever  the  force  may  be,  it  is  one 
which  acts  normally  to  the  surface  of  the  ovum,  and  after  entry  the 

*  Cf.  N.  Rashevsky,  Mathematical  Biophysics,  Chicago,  1938;  and  many  earlier 
papers.  Eg.  Physico-matheraatical  aspects  of  cellular  multiplication  and  de- 
velopment, Cold  Spring  Harbor  Symposia,  ii,  1934;  The  mechanism  of  division  of 
small  liquid  systems  which  are  the  seat  of  physico-chemical  reactions,  Physics,  in, 
pp.  374-379,  1934;    papers  in  Protoplasma,  xiv-xx,  1931-33,  etc. 

t  With  the  classical  account  by  H.  Fol,  C.R.  lxxxiii,  p.  667,  1876;  Mem.  Soc. 
Phys.  Geneve,  xxvi,  p.  89,  1879,  cf.  Robert  Chambers,  The  mechanism  of  the  entrance 
of  sperm  into  the  star-fish  egg,  Journ.  Gen,  Physiol,  v,  pp.  821-829,  1923.  Here 
a  delicate  filament  is  said  to  run  out  from  the  fertilisation -cone  and  drag  the 
spermatozoon  in;  but  this  is  disputed  and  denied  by  E.  Just,  Biol.  Bull.  LVii, 
pp.  311-325,  1929. 

I  But,  under  artificial  conditions,  "polyspermy"  may  take  place,  eg.  under 
the  action  of  dilute  poisons,  or  of  an  abnormally  high  temperature,  these  being 
doubtless  also  conditions  under  which  the  surface-tension  is  diminished. 


336  ON  THE  INTERNAL  FORM  [ch. 

spermatozoon  points  straight  towards  the  centre  of  the  egg.  From 
the  fact  that  other  spermatozoa,  subsequent  to  the  first,  fail  to 
effect  an  entry,  we  may  safely  conclude  that  an  immediate  con- 
sequence of  the  entry  of  the  spermatozoon  is  an  increase  in  the 
surface-tension  of  the  egg:  this  being  but  one  of  the  complex 
reactions  exhibited  by  the  surface,  or  cortex  of  the  cell*.  Some- 
where or  other,  within  the  egg,  near  or  far  away,  lies  its  own  nuclear 
body,  the  so-called  female  pronucleus,  and  we  find  that  after  a 
while  this  has  fused  with  the  "male  pronucleus"  or  head  of  the 
spermatozoon,  and  that,  the  body  resulting  from  their  fusion  has 
come  to  occupy  the  centre  of  the  egg.  This  must  be  due  (as  Whitman 
pointed  out  many  years  ago)  to  a  force  of  attraction  acting  between 
the  two  bodies,  and  another  force  acting  upon  one  or  other  or  both 
in  the  direction  of  the  centre  of  the  cell.  Did  we  know  the  magnitude 
of  these  several  forces,  it  would  be  an  easy  task  to  calculate  the 
precise  path  which  the  two  pronuclei  would  follow,  leading  to  con- 
jugation and  to  the  central  position.  As  we  do  not  know  the 
magnitude,  but  only  the  direction,  of  these  forces,  we  can  only  make 
a  general  statement:  (1)  the  paths  of  both  moving  bodies  will  he 
wholly  within  a  plane  triangle  drawn  between  the  two  bodies  and 
the  centre  of  the  cell;  (2)  unless  the  two  bodies  happen  to  he,  to 
begin  with,  precisely  on  a  diameter  of  the  cell,  their  paths  until  they 
meet  one  another  will  be  curved  paths,  the  convexity  of  the  curve 
being  towards  the  straight  line  joining  the  two  bodies;  (3)  the  two 
bodies  will  meet  a  httle  before  they  reach  the  centre;  and,  having 
met  and  fused,  will  travel  on  to  reach  the  centre  in  a  straight  hne. 
The  actual  study  and  observation  of  the  path  followed  is  not  very 
easy,  owing  to  the  fact  that  what  we  usually  see  is  not  the  path 
itself,  but  only  a  projection  of  the  path  upon  the  plane  of  the 
microscope ;  but  the  curved  path  is  particularly  well  seen  in  the  frog's 
egg,  where  the  path  of  the  spermatozoon  is  marked  by  a  little  streak 
of  brown  pigment,  and  the  fact  of  the  meeting  of  the  pronuclei  before 
reaching  the  centre  has  been  repeatedly  seen  by  many  observers  f. 

*  See  Mrs  Andrews'  beautiful  observations  on  "Some  spinning  activities  of 
protoplasm  in  starfish  and  echinoid  eggs,"  J  own.  Morphol.  xii,  pp.  307-389,  1897. 

t  W.  Pfeffer,  Locomotorische  Richtungsbewegungen  durch  chemische  Reize, 
Unters.  a.  d.  Botan.  Inst.  Tubingen,  i,  1884;  Physiology  of  Plants,  m,  p.  345,  Oxford, 
1906;  W.  J.  Dakin  and  M.  G.  C,  Fordham,  Journ.  Exp.  Biol.  t.  pp.  183-200,  1924. 
Cf.  J.  Loeb,  Dynamics  of  Living  Matter,  1906,  p.  153. 


IV]  AND  STRUCTURE  OF  THE  CELL  337 

The  problem  recalls  the  famous  problem  of  three  bodies,  which  has 
so  occupied  the  astronomers;  and  it  is  obvious  that  the  foregoing 
brief  description  is  very  far  from  including  all  possible  cases. 
Many  of  these  are  particularly  described  in  the  works  of  Fol,  Roux, 
Whitman  and  others*. 

The  intracellular  phenomena  of  which  we  have  now  spoken  have 
assumed  great  importance  in  biological  literature  and  discussion 
during  the  last  fifty  years;  but  it  is  open  to  us  to  doubt  whether 
they  will  be  found  in  the  end  to  possess  more  than  a  secondary, 
even  a  remote,  biological  significance.  Most,  if  not  all  of  them, 
would  seem  to  follow  immediately  and  inevitably  from  certain 
simple  assumptions  as  to  the  physical  constitution  of  the  cell,  and 
from  an  extremely  simple  distribution  of  polarised  forces  within  it. 
We  have  already  seen  that  how  a  thing  grows,  and  what  it  grows 
into,  is  a  dynamic  and  not  a  merely  material  problem;  so  far  as 
the  material  substance  is  concerned,  it  is  so  only  by  reason  of  the 
chemical,  electrical  or  other  forces  which  are  associated  with  it. 
But  there  is  another  consideration  which  would  lead  us  to  suspect 
that  many  features  in  the  structure  and  configuration  of  the  cell 
are  of  secondary  biological  importance;  and  that  is,  the  great 
variation  to  which  these  phenomena  are  subject  in  similar  or  closely 
related  organisms,  and  the  apparent  impossibihty  of  correlating 
them  with  the  pecuHarities  of  the  organism  as  a  whole.  In  a 
broad  and  general  way  the  phenomena  are  always  the  same.  Certain 
structures  swell  and  contract,  twine  and  untwine,  split  and  unite, 
advance  and  retire ;  certain  chemical  changes  also  repeat  themselves. 
But  Nature  rings  the  changes  on  all  the  details.  "Comparative 
study  has  shewn  that  almost  every  detail  of  the  processes  (of 
mitosis)  described  above  is  subject  to  variation  in  different  forms 
of  cells  |."  A  multitude  of  cells  divide  to  the  accompaniment  of 
caryokinetic  phenomena;  but  others  do  so  without  any  visible 
caryokinesis  at  all.     Sometimes  the  polarised  field  of  force  is  within, 

*  H.  Fol,  Becherches  sur  la  fecondation,  1879;  W.  Roux,  Beitrage  zur  Erit- 
wickelungsmechanik  des  Embryos,  Arch.  f.  Mikr.  Anat.  xix,  1887;  C.  0.  Whitman, 
Ookinesis,  Journ.  Morph.  i,  1887;  E.  Giglio-Tos,  Entwicklungsmechanische 
Studien,  I,  Arch.  f.  Entw.  Mech.  li,  p.  94,  1922,  See  also  Frank  R.  Lillie,  Problems 
of  Fertilisation,  Chicago,  1919. 

t  Wilson,  The  Cell.  p.  77;  cf.  3rd  ed.  (1925),  p.  120. 


338        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

sometimes  it  is  adjacent  to,  and  at  other  times  it  lies  remote  from, 
the  nucleus.  The  distribution  of  potential  is  very  often  symmetrical 
and  bipolar,  as  in  the  case  described;  but  a  less  symmetrical 
distribution  often  occurs,  with  the  result  that  we  have,  for  a  time 
at  least,  numerous  centres  of  force,  instead  of  the  two  main  correlated 
poles :  this  is  the  simple  explanation  of  the  numerous  stellate  figures, 


Haploid  number  of  chr.'raosomes 
Fig.  103.    Summation  diagram  shewing  the  %  number  of  instances  (among  2,415 
phanerogams  and  1,070  metazoa),  in  which  the  chromosomes  do  not  exceed 
a  given  number.    Data  from  M.  J.  D.  White. 

or  "Strahlungen,"  which  have  been  described  in  certain  eggs,  such  as 
those  of  Chaetopterus.  The  number  of  chromosomes  may  be  constant 
within  a  group,  as  in  the  tailed  Amphibia,  with  12;  or  very  variable, 
as  in  sedges,  and  in  grasshoppers  * ;  in  one  and  the  same  species 
of  worm  (Ascaris  megdlocephala),  one  group  or  two  groups  of 
chromosomes  may  be  present.  And  remarkably  constant,  in 
general,  as  the  number  in  any  one  species  undoubtedly  is,  yet  we 
must  not  forget  that,  in  plants  and  animals  alike,  the  whole  range 
of  observed  numbers  is  but  a  small  one  (Fig.  103);    for  (as  regards 

*  There  are  varieties  of  Artemia  salina  which  hardly  differ  in  outward  characters, 
but  differ  widely  in  the  number  of  their  chromosomes. 


IV]  OF  CHROMOSOMES  339 

the  germ-nuclei)  few  have  less  than  six  chromosomes,  and  few  have 
more  than  twenty*.  In  closely  related  animals,  such  as  various 
species  of  Copepods,  and  even  in  the  same  species  of  worm  or  insect, 
the  form  of  the  chromosomes  and  their  arrangement  in  relation*  to 
the  nuclear  spindle  have  been  found  to  differ  in  ways  alluded  to 
above ;  while  only  here  and  there,  as  among  the  chrysanthemums, 
do  related  species  or  varieties  shew  their  own  characteristic  chromo- 
some numbers.  In  contrast  to  the  narrow  range  of  the  chromo- 
some numbers,  we  may  reflect  on  the  all  but  infinite  possibilities  of 
chemical  variabihty.  Miescher  shewed  that  a  molecule  containing 
40  C-atoms  would  admit  (arithmetically  though  not  necessarily 
chemically)  of  a  million  possible  isomers;  and  changes  in  position 
of  the  N-atoms  of  a  protein,  for  instance,  might  vastly  increase 
that  prodigious  number.  In  short,  we  cannot  help  perceiving 
that  many  nuclear  phenomena  are  not  specifically  related  to  the 
particular  organism  in  which  they  have  been  observed,  and  that 
some  are  not  even  specially  and  indisputably  connected  with  the 
organism  as  such.  They  include  such  manifestations  of  the  physical 
forces,  in  their  various  permutations  and  combinations,  as  may  also 
be  witnessed,  under  appropriate  conditions,  in  non-living  things. 
When  we  attempt  to  separate  our  purely  morphological  or  "purely 

*  The  commonest  numbers  of  (haploid)  chromosomes,  both  in  plants  and 
animals,  are  8,  12  and  16.  The  median  number  is  12  in  both,  and  the  lower 
quartile  is  8,  likewise  in  both;  but  the  upper  quartile  is  24  or  thereby  in  animals, 
and  in  the  neighbourhood  of  16  in  plants.  If  we  may  judge  by  the  long  lists  given 
by  E.  B.  Wilson  (The  Cell,  3rd  ed.  pp.  855-865),  by  M.  Ishikawa  in  Botan.  Mag. 
Tokyo,  XXX,  1916,  by  M.  J.  D.  White  in  his  book  on  Chromosomes,  or  by  Tischler 
in  Tabulae  Biologicae  (1927),  fully  60  per  cent,  of  the  observed  cases  lie  between  6 
and  16.  As  Wilson  says  (p.  866)  "the  number  of  chromosomes  is  per  se  a  matter 
of  secondary  importance";  and  (p.  868)  "We  must  admit  the  present  inadequacy 
of  attempts  to  reduce  the  chromosome  numbers  to  any  single  or  consistent 
arithmetical  rules."  Clifford  Dobell  had  said  the  same  thing:  "Nobody  nowadays 
will  be  prepared  to  argue  that  chromosome  numbers,  as  such,  have  any  quantitative 
or  qualitative  relation  to  the  characters  exhibited  by  their  owners.  Complexity 
of  bodily  structure  is  certainly  not  correlated  in  any  way  with  multiplicity  of 
chromosomes  " ;  La  Cellule,  xxxv,  p.  188,  1924.  On  the  other  hand,  Tischler  stoutly 
maintains  that  chromosome-numbers  give  useful  evidence  of  phylogenetic  affinity 
{Biol.  Centralbl.  XLvni,  pp.  321-345,  1928);  and  there  axe  a  few  well-known  cases, 
such  as  the  chrysanthemums,  where,  undoubtedly,  the  numbers  are  constant  and 
specific.  Again  in  certain  cases,  the  number  of  the  chromosomes  may- differ  in 
dififerent  races  (diploid  and  tetraploid)  of  the  same  plant;  and  the  difference  is 
accompanied  by  differences  in  cell-size,  in  rate  of  growth,  and  even  in  the  shape 
of  the  fruit  (of.  Sinnott  and  Blakeslee,  Xat.  Acad,  of  Sci.  1938,  p.  476). 


340        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

embryological "  studies  from  physiological  and  physical  investiga- 
tions, we  tend  ipso  facto  to  regard  each  particular  structure  and 
configuration  as  an  attribute,  or  a  particular  "character,"  of  this  or 
that  particular  organism.  From  this  assumption  we  are  easily  led  to 
the  framing  of  theories  as  to  the  ancestral  history,  the  classificatory 
position,  the  natural  affinities  of  the  several  organisms :  in  fact,  to 
apply  our  embryological  knowledge  to  the  study  of  phylogeny. 
When  we  find,  as  we  are  not  long  of  finding,  that  our  phylogenetic 
hypotheses  become  complex  and  unwieldy,  we  are  nevertheless 
reluctant  to  admit  that  the  whole  method,  with  its  fundamental 
postulates,  is  at  fault;  and  yet  nothing  short  of  this  would  seem 
to  be  the  case,  in  regard  to  the  earher  phases  at  least  of  embryonic 
development.  All  the  evidence  at  hand  goes,  as  it  seems  to  me,  to 
shew  that  embryological  data,  prior  to  and  even  long  after  the 
epoch  of  segmentation,  are  essentially  a  subject  for  physiological  and 
physical  investigation  and  have  but  the  shghtest  fink,  if  any,  with 
the  problems  of  zoological  classification.  Comparative  embryology 
has  its  own  facts  to  classify,  and  its  own  methods  and  principles  of 
classification.  We  may  classify  eggs  according  to  the  presence  or 
absence,  the  paucity  or  abundance,  of  their  associated  food-yolk, 
the  chromosomes  according  to  their  form  and  their  number,  the 
segmentation  according  to  its  various  "types" — radial,  bilateral, 
spiral,  and  so  forth.  But  we  have  httle  right  to  expect,  and  in 
point  of  fact  we  shall  very  seldom  and  (as  it  were)  only  accidentally 
find,  that  these  embryological  categories  coincide  with  the  lines  of 
"natural"  or  "phylogenetic"  classification  which  have  been  arrived 
at  by  the  systematic  zoologist. 

The  efforts  to  explain  "heredity"  by  help  of  "genes"  and  chromo- 
somes, which  have  grown  up  in  the  hands  of  Morgan  and  others  since 
this  book  was  first  written,  stand  by  themselves  in  a  category  which 
is  all  their  own  and  constitutes  a  science  which  is  justified  of 
itself.  To  weigh  or  criticise  these  explanations  would  lie  outside 
my  purpose,  even  were  I  fitted  to  attempt  the  task.  When  these 
great  discoveries  began  to  be  made,  Bateson  crossed  the  ocean 
to  see  and  hear  for  himself  what  Morgan  and  his  pupils  had  to 
shew  and  to  tell.  He  came  home  convinced,  and  humbly  marvelling. 
And  I  leave  this  great  subject  on  one  side  not  because  I  doubt  for  a 
moment  the  facts  nor  dispute  the  hypotheses  nor  decry  the  im- 


IV]  OF  THE  CELL-THEORY  341 

portance  of  one  or  other;  but  because  we  are  so  much  in  the  dark 
as  to  the  mysterious  field  of  force  in  which  the  chromosomes  he, 
far  from  the  visible  horizon  of  physical  science,  that  the  matter  lies 
(for  the  present)  beyond  the  range  of  problems  which  this  book 
professes  to  discuss,  and  the  trend  of  reasoning  which  it  endeavours 
to  mamtain. 

The  cell*,  which  Goodsir  spoke  of  as  a  centre  of  force,  is  in  reality 
a  sphere  of  action  of  certain  more  or  less  localised  forces;  and  of 
these,  surface-tension  is  the  particular  force  which  is  especially 
responsible  for  giving  to  the  cell  its  outline  and  its  morphological 
individuahty.  The  partially  segmented  differs  from  the  totally 
segmented  egg,  the  unicellular  Infusorian  from  the  minute  multi- 

*  The  "  cell -theory "  began  early  and  grew  slowly.  In  a  curious  passage  which 
Mr  Clifford  Dobell  has  shewn  me  {Nov.  Org.  ii,  7,  ad  fin.).  Bacon  speaks  of  "cells" 
in  the  human  body:  of  a  "  coUocatio  spiritus  per  corpoream  molem,  eiusque  pori, 
meatus,  venae  et  cellulae,  et  rudimenta  sive  tentamenta  corporis  organici."  It  is 
"  surely  one  of  the  most  strangely  prophetic  utterances  which  even  Bacon  ever 
made."  Apart  from  this  the  story  begins  in  the  seventeenth  century,  with  Robert 
Hooke's  well-known  figure  of  the  "cells"  in  a  piece  of  cork  (1665),  with  Grew's 
"bladders"  or  "bubbles"  in  the  parenchyma  of  young  beans,  and  Malpighi's 
"utriculi"  or  "sacculi"  in  the  parenchyma  or  "utriculorum  substantia"  of 
various  plants.  Christian  Fr.  v.  Wolff  conceived,  about  the  same  time,  a  hypo- 
thetical "cell-theory,"  on  the  analogy  of  Leibniz's  Monads;  but  the  first  clear 
idea  of  a  cellular  parenchyma,  or  contextus  cellularis,  came  from  C.  Gottlieb 
Ludwig  (1742),  and  from  K.  Fr.  Wolff,  who  spoke  freely  of  cells  or  cellulae. 
Fontana,  author  of  a  curious  Traite  sur  le  venin  de  la  vipere  (1781),  described 
various  histological  elements,  caught  a  glimpse  of  the  nucleus,  and  experi- 
mented with  reagents,  using  syrup  of  violets  for  a  stain.  Early  in  the 
eighteenth  century  the  vessels  of  the  plant  played  an  important  role,  under  Kurt 
Sprengel  and  Treviranus;  but  it  was  not  till  1831  that  Hugo  v.  Mohl  recognised 
that  they  also  arose  from  "cells."  About  this  time  Robert  Brown  discovered, 
or  re-discovered,  the  nucleus  (1833),  which  Schleiden  called  the  cytohlast,  or  "cell- 
producer."  It  was  Schleiden's  idea,  and  a  far-seeing  one,  that  the  cell  lived  a  double 
life,  a  life  of  its  own  and  the  life  of  the  plant  to  which  it  belonged:  "jede  Zelle 
fuhrt  nun  ein  zweifaches  Leben :  ein  selbststandiges,  nur  ihrer  eigenen  Entwicklung 
angehorigen,  und  ein  anderes  mittelbares,  insofern  sie  integrierender  Theil  einer 
Pfianze  geworden  ist "  {Phylogenesis,  1838,  p.  1 ).  The  cell-theory,  so  long  a- building, 
may  be  said  to  have  been  launched,  and  christened,  with  Schwann's  Mikroskopische 
Untersmhungen  of  1839.  Within  the  next  five  years  Martin  Barry  shewed  how 
cell-division  starts  with  the  nucleus,  Henle  described  the  budding  of  certain  cells, 
and  Goodsir  declared  that  all  cells  originate  in  pre-existing  cells,  a  doctrine  at  once 
accepted  by  Remak,  and  made  famous  in  pathology  by  Virchow.  (Cf.  {int.  al.) 
J.  G.  McKendrick,  On  the  modern  cell-theory,  etc.,  Proc  Phil.  Soc.  Glasgow,  xix, 
pp.  1-55,  1887;  J.  Stephenson,  Robert  Brown. .  .and  the  cell-theory,  Proc.  Linn. 
Soc.  1931-2,  pp.  45-54;  M.  Mobius,  Hundert  Jahre  Zellenlehre,  Jen.  Ztschr.  lxxi, 
pp.  313-326,  1938.) 


342        FORM  AND  STRUCTURE  OF  THE  CELL       [ch. 

cellular  Turbellarian,  in  the  intensity  and  the  range  of  those  surface- 
tensions  which  in  the  one  case  succeed  and  in  the  other  fail  to  form 
a  visible  separation  between  the  cells.  Adam  Sedgwick  used  to 
call  attention  to  the  fact  that  very  often,  even  in  eggs  that  appear 
to  be  totally  segmented,  it  is  yet  impossible  to  discover  an  actual 
separation  or  cleavage,  through  and  through,  between  the  cells  which 
on  the  surface  of  the  egg  are  so  clearly  delimited;  so  far  and  no 
farther  have  the  physical  forces  effectuated  a  visible  "cleavage." 
The  vacuolation  of  the  protoplasm  in  Actinophrys  or  Actinosphaerium 
is  due  to  localised  surface-tensions,  quite  irrespective  of  the  multi- 
nuclear  nature  of  the  latter  organism.  In  short,  the  boundary  walls 
due  to  surface-tension  may  be  present  or  may  be  absent,  with  or 
without  the  delimination  of  the  other  specific  fields  of  force  which 
are  usually  correlated  with  these  boundaries  and  with  the  inde- 
pendent individuality  of  the  cells.  What  we  may  safely  admit, 
however,  is  that  one  effect  of  these  circumscribed  fields  of  force  is 
usually  such  a  separation  or  segregation  of  the  protoplasmic 
constituents,  the  more  fluid  from  the  less  fluid  and  so  forth,  as  to 
give  a  field  where  surface-tension  may  do  its  work  and  bring  a 
visible  bouhdary  into  being.  When  the  formation  of  a  "surface" 
is  once  effected,  its  physical  condition,  or  phase,  will  be  bound  to 
differ  notably  from  that  of  the  interior  of  the  cell,  and  under 
appropriate  chemical  conditions  the  formation  of  an  actual  cell- wall, 
cellulose  or  other,  is  easily  inteUigible.  To  this  subject  we  shall 
return  again,  in  another  chapter. 

From  the  moment  that  we  enter  on  a  dynamical  conception  of 
the  cell,  we  perceive  that  the  old  debates  were  vain  as  to  what 
visible  portions  of  the  cell  were  active  or  passive,  living  or  non- 
living. For  the  manifestations  of  force  can  only  be  due  to  the 
interaction  of  the  various  parts,  to  the  transference  of  energy  from 
one  to  another.  Certain  properties  may  be  manifested,  certain 
functions  may  be  carried  on,  by  the  protoplasm  apart  from  the 
nucleus;  but  the  interaction  of  the  two  is  necessary,  that  other 
and  more  important  properties  or  functions  may  be  manifested. 
We  know,  for  instance,  that  portions  of  an  Infusorian  are  incapable 
of  regenerating  lost  parts  in  the  absence  of  a  nucleus,  while  nucleated 
pieces  soon  regain  the  specific  form  of  the  organism:  and  we  are 
told    that    reproduction    by    fission    cannot    be    initiated,    though 


IV]  OF  THE  CELL-THEORY  343 

apparently  all  its  later  steps  can  be  carried  on,  independently  of 
nuclear  action.  Nor,  as  Verworn  pointed  out,  can  the  nucleus 
possibly  be  regarded  as  the  "sole  vehicle  of  inheritance,"  since  only 
in  the  conjunction  of  cell  and  nucleus  do  we  find  the  essentials  of 
cell-hfe.  "Kern  und  Protoplasma  sind  nur  vereint  lebensfahig,"  as 
Nussbaum  said.  Indeed  we  may,  with  E.  B.  Wilson,  go  further, 
and  say  that  "the  terms  'nucleus'  and  'cell-body'  should  probably 
be  regarded  as  only  topographical  expressions  denoting  two 
differentiated  areas  in  a  common  structural  basis." 

Endless  discussion  has  taken  place  regarding  the  centrosome, 
some  holding  that  it  is  a  specific  and  essential  structure,  a  permanent 
corpuscle  derived  from  a  similar  pre-existing  corpuscle,  a  "fertilising 
element"  in  the  spermatozoon,  a  special  "organ  of  cell-division," 
a  material  "dynamic  centre"  of  the  cell  (as  Van  Beneden  and 
Boveri  call  it);  while  on  the  other  hand,  it  is  pointed  out  that 
many  cells  live  and  multiply  without  any  visible  centrosomes,  that 
a  centrosome  may  disappear  and  be  created  anew,  and  even  that 
under  artificial  conditions  abnormal  chemical  stimuh  may  lead  to 
the  formation  of  new  centrosomes.  We  may  safely  take  it  that  the 
centrosome,  or  the  "attraction  sphere,"  is  essentially  a  "centre  of 
force,"  and  that  this  dynamic  centre  may  or  may  not  be  constituted 
by  (but  will  be  very  apt  to  produce)  a  concrete  and  visible  con- 
centration of  matter. 

It  is  far  from  correct  to  say,  as  is  often  done,  that  the  cell- wall, 
or  cell-membrane,  belongs  "to  the  passive  products  of  protoplasm 
rather  than  to  the  hving  cell  itself";  or  to  say  that  in  the  animal 
cell,  the  cell-wall,  because  it  is  "slightly  developed,"  is  relatively 
unimportant  compared  with  the  important  role  which  it  assumes 
in  plants.  On  the  contrary,  it  is  quite  certain  that,  whether  visibly 
diiferentiated  into  a  semi-permeable  membrane  or  merely  con- 
stituted by  a  liquid  film,  the  surface  of  the  cell  is  the  seat  of 
important  forces,  capillary  and  electrical,  which  play  an  essential 
part  in  the  dynamics  of  the  cell.  Even  in  the  thickened,  largely 
solidified  cellulose  wall  of  the  plant-cell,  apart  from  the  mechanical 
resistances  which  it  affords,  the  osmotic  forces  developed  in  con- 
nection with  it  are  of  essential  importance. 

But  if  the  cell  acts,  after  this  fashion,  as  a  whole,  each  part 
interacting  of  necessity  with  the  rest,  the  same  is  certainly  true  of 


344   FORM  AND  STRUCTURE  OF  THE  CELL   [ch. 

the  entire  multicellular  organism :  as  Schwann  said  of  old,  in  very- 
precise  and  adequate  words,  "the  whole  organism  subsists  only  by 
means  of  the  recijpTocal  action  of  the  single  elementary  parts*."  As 
Wilson  says  again,  "the  physiological  autonomy  of  the  individual 
cell  falls  into  the  background . . .  and  the  apparently  composite 
character  which  the  multicellular  organism  may  exhibit  is  owing  to 
a  secondary  distribution  of  its  energies  among  local  centres  of 
action!."  I*  is  here  that  the  homology  breaks  down  which  is  so 
often  drawn,  and  overdrawn,  between  the  unicellular  organism  and 
the  individual  cell  of  the  metazoonj. 

Whitman,  Adam  Sedgwick  §,  and  others  have  lost  no  opportunity 
of  warning  us  against  a  too  hteral  acceptation  of  the  cell-theory, 
against  the  view  that  the  multicellular  organism  is  a  colony  (or,  as 
Haeckel  called  it,  in  the  case  of  the  plant,  a  "republic")  of  inde- 
pendent units  of  Ufe||.  As  Goethe  said  long  ago,  "Das  lebendige 
ist  zwar  in  Elemente  zerlegt,  aber  man  kann  es  aus  diesen  nicht 
wieder  zusammenstellen  und  beleben " ;  the  dictum  of  the  Cellular- 
pathologie  being  just  the  opposite,  "Jedes  Thier  erscheint  als  cine 
Summe  vitaler  Einheiten,  von  denen  jede  den  vollen  Charakter  des 
Lebens  an  sich  trdgt." 

Hofmeister  and  Sachs  have  taught  us  that  in  the  plant  the  growth 

♦  Theory  of  Cells,  p.  191. 

t  The  Cell  in  Development,  etc.,  p.  59;   cf.  3rd  ed.  (1925),  p.  102.- 

X  E.g.  Brticke,  Elementarorganismen,  p.  387:  "Wir  miissen  in  der  Zelle  einen 
kleinen  Thierleib  sehen,  und  diirfen  die  Analogien,  welche  zwischen  ihr  und  den 
kleinsten  Thierformen  existiren,  niemals  aus  den  Augen  lassen." 

§  C.  0.  Whitman,  The  inadequacy  of  the  cell-theory,  Journ.  Morphol.  viii, 
pp.  639-658,  1893;  A.  Sedgwick,  On  the  inadequacy  of  the  cellular  theory  of 
development,  Q.J. M.S.  xxxvii,  pp.  87-101,  1895;  xxxviii,  pp.  331-337,  1896. 
Cf.  G.  C.  Bourne,  ibid,  xxxviii,  pp.  137-174,  1896;  Clifford  Dobell,  The  principles 
of  Protistology,  Arch.  f.  Protistenk.  xxiii,  p.  270,  1911. 

II  Cf.  0.  Hertwig,  Die  Zelle  und  die  Gewebe,  1893,  p.  1:  "Die  Zellen,  in  welche 
der  Anatom  die  pflanzlichen  und  thierischen  Organismen  zerlegt,  sind  die  Trager 
der  Lebensfunktionen ;  sie  sind,  wie  Virchow  sich  ausgedruckt  hat,  die  'Lebensein- 
heiten.'  Von  diesem  Gesichtspunkt  aus  betrachtet,  erscheint  der  Gesammtlebens- 
prozess  eines  zusammengesetzten  Organismus  nichts  Anderes  zu  sein  als  das  hochst 
yerwickelte  Resultat  der  einzelnen  Lebensprozesse  seiner  zahlreichen,  verschieden 
functionirenden  Zellen."  But  in  1920  Doncaster  (Cytology,  p.  1)  declared  that  "the 
old  idea  of  discrete  and  independent  cells  is  almost  abandoned,"  and  that  the 
word  cell  was  coming  to  be  used  "rather  as  a  convenient  descriptive  term  than 
as  denoting  a  fundamental  concept  of  biology";  and  James  Gray  {Experimental 
Cytology,  p.  2)  said,  in  1931,  that  "we  must  be  careful  to  avoid  any  tacit  assumption 
that  the  cell  is  a  natural,  or  even  legitimate,  unit  of  life  and  function." 


IV]  OF  THE  CELL-THEORY  345 

of  the  mass,  the  growth  of  the  organ,  is  the  primary  fact,  that 
"cell  formation  is  a  phenomenon  very  general  in  organic  life,  but 
still  only  of  secondary  significance."  "Comparative  embryology,'* 
says  Whitman,  "reminds  us  at  every  turn  that  the  organism 
dominates  cell-formation,  using  for  the  same  purpose  one,  several, 
or  many  cells,  massing  its  material  and  directing  its  movements 
and  shaping  its  organs,  as  if  cells  did  not  exist*."  So  Rauber 
declared  that,  in  the  whole  world  of  organisms,  "das  Ganze  liefert 
die  Theile,  nicht  die  Theile  das  Ganze:  letzteres  setzt  die  Theile 
zusammen,  nicht  diese  jenesf."  And  on  the  botanical  side  De  Bary 
has  summed  up  the  matter  in  an  aphorism,  "Die  Pflanze  bildet 
Zellen,  nicht  die  Zelle  bildet  Pflanzen." 

Discussed  almost  wholly  from  the  concrete,  or  morphological 
point  of  view,  the  question  has  for  the  most  part  been  made  to  turn 
on  whether  actual  protoplasmic  continuity  can  be  demonstrated 
between  one  cell  and  another,  whether  the  organism  be  an  actual 
reticulum,  or  syncytium  J.  But  from  the  dynamical  point  of  view 
the  question  is  much  simpler.  We  then  deal  not  with  material 
continuity,  not  with  little  bridges  of  connecting  protoplasm,  but 
with  a  continuity  of  forces,  a  comprehensive  field  of  force,  which 
runs  through  and*  through  the  entire  organism  and  is  by  no  means 
restricted  in  its  passage  to  a  protoplasmic  continuum.  And  such 
a  continuous  field  of  force,  somehow  shaping  the  whole  organism, 
independently  of  the  number,  magnitude  and  form  of  the  individual 
cells,  which  enter  hke  a  froth  into  its  fabric,  seems  to  me  certainly 
and  obviously  to  exist.  As  Whitman  says,  "the  fact  that  physio- 
logical unity  is  not  broken  by  cell-boundaries  is  confirmed  in  so 
many  ways  that  it  must  be  accepted  as  one  of  the  fundamental 
truths  of  biology  §." 

*  Journ.  Morph.  viii,  p.  653,  1893. 

t  Neue  Grundlegungen  zur  Kenntniss  der  Zelle,  Morph.  Jahrb.  viii,  pp.  272, 
313,  333,  1883. 

X  Cf.  e.g.  Ch.  van  Bambeke,  A  propos  de  la  delimitation  cellulaire,  Bull.  Soc. 
beige  de  Microsc.  xxiii,  pp.  72-87,  1897. 

§  Journ.  Morph.  ii,  p.  49,  1889. 


CHAPTER  V 

THE   FORMS   OF  CELLS 

Protoplasm,  as  we  have  already  said,  is  a  fluid*  or  a  semi-fluid 
substance,  and  we  need  not  try  to  describe  the  particular  properties 
of  the  colloid  or  jelly-like 'substances  to  which  it  is  alHed,  or  rather 
the  characteristics  of  the  "colloidal  state"  in  which  it  and  they 
exist;  we  should  find  it  no  easy  matter f.  Nor  need  we  appeal  to 
precise  theoretical  definitions  of  fluidity,  lest  we  come  into  a 
debatable  land.  It  is  in  the  most  general  sense  that  protoplasm 
is  "fluid."  As  Graham  said  (of  colloid  matter  in  general),  "its 
softness  partakes  of  fluidity,  and  enables  the  colloid  to  become  a 
vehicle  for  liquid  diffusion,  like  water  itself  J."  When  we  can  deal 
with  protoplasm  in  sufficient  quantity  we  see  it /oi^§;  particles 
move  freely  through  it,  air-bubbles  and  liquid  droplets  shew  round 
or  spherical  within  it;  and  we  shall  have  much  to  say  about  other 
phenomena  manifested  by  its  own  surface,  which  are  those  especially 
characteristic  of  liquids.  It  may  encompass  and  contain  solid 
bodies,  and  it  may  "secrete"  solid  substances  within  or  around 
itself;  and  it  often  happens  in  the  complex  Hving  organism  that 
these  sohd  substances,  such  as  shell  or  nail  or  horn  or  feather, 
remain  when  the  protoplasm  which  formed  them  is  dead  and  gone. 
But  the  protoplasm  itself  is  fluid  or  semi-fluid,  and  permits  of  free 
(though  not  necessarily  rapid)  diffusion  and  easy  convection  of 
particles  within  itself,  which  simple  fact  is  of  elementary  importance 

*  Cf.  W.  Kuhne,  Ueber  das  Protoplasma,  1864. 

t  Sand,  or  a  heap  of  millet-seed,  may  in  a  sense  be  deemed  a  "fluid,"  and  such 
the  learned  Father  Boscovich  held  them  to  be  {Theoria,  p.  427),  but  at  best  they 
are  fluids  without  a  surface.  Galileo  had  drawn  the  same  comparison;  but  went  on 
to  contrast  the  continuity,  or  infinite  subdivision,  of  a  fluid  with  the  finite,  dis- 
continuous subdivision  of  a  fine  powder.  Cf.  Boyer,  Concepts  of  the  Calculus,  1939, 
p.  291.  . 

J  Phil  Trans,  clt,  p  183,  1861;  Researches,  ed.  Angus  Smith,  1877,  p  553. 
We  no  longer  speak,  however,  of  "colloids'  in  a  specific  sense,  as  Graham  did; 
for  any  substance  can  be  brought  into  the  "colloidal  state"  by  appropriate  means 
or  in  an  appropriate  medmm. 

§  The  copious  protoplasm  of  a  Myxomycete  has  been  passed  unharmed  through 
filter-paper  with  a  pore-size  of  about  1  /x,  or  0001  mm. 


CH.  V]  OF  THE  COLLOID  STATE  347 

in  connection  with  form,  throwing  light  on  what  seem  to  be  common 
characteristics  and  pecuHarities  of  the  forms  of  Hving  things. 

Much  has  been  done,  and  more  said,  about  the  nature  of  protoplasm  since 
this  book  was  written.  Calling  cytoplasm  the  cell-protoplasm  after  deduction 
of  chloroplasts  and  other  gross  inclusions,  we  find  it  to  contain  fats,  proteins, 
lecithin  and  some  other  substances  combined  with  much  water  (up  to 
97  per  cent.)  to  form  a  sort  of  watery  gel.  The  microscopic  structures 
attributed  to  it,  alveolar,  granular  or  fibrillar,  are  inconstant  or  invalid; 
but  it  does  appear  to  possess  an  invisible  or  submicroscopic  structure, 
distinguishing  it  from  an  ordinary  colloid  gel,  and  forming  a  quasi-solid 
framework  or  reticulum.  This  framework  is  based  on  proteid  macromolecules, 
in  the  form  of  polypeptide  chains,  of  great  length  and  carrying  in  side-chains 
other  organic  constituents  of  the  cytoplasm*.  The  polymerised  units 
represent  the  micellae  f  which  the  genius  of  Nageli  predicted  or  postulated 
more  than  sixty  years  ago;  and  we  may  speak  of  a  "micellar  framework" 
as  representing  in  our  cytoplasm  the  dispersed  phase  of  an  ordinary  colloid. 
In  short,  as  the  cytoplasm  is  neither  true  fluid  not  true  solid,  neither  is  it  true 
colloid  in  the  ordinary  sense.  Its  micellar  structure  gives  it  a  certain  rigidity 
or  tendency  to  retain  its  shape,  a  certain  plasticity  and  tensile  strength,  a 
certain  ductility  or  capacity  to  be  drawn  out  in  threads;  but  yet  leaves  it 
with  a  permeability  (or  semi-permeability),  a  capacity  to  swell  by  imbibition, 
above  all  an  ability  to  stream  and  flow,  which  justify  our  calling  it  "fluid 
or  semi-fluid,"  and  account  for  its  exhibition  of  surface-tension  and  other 
capillary  phenomena. 

The  older  naturalists,  in  discussing  the  differences  between  organic  and 
inorganic  bodies,  laid  stress  upon  the  circumstance  that  the  latter  grow  by 
"agglutination,"  and  the  former  by  what  they  termed  "intussusception." 
The  contrast  is  true;  but  it  applies  rather  to  solid  or  crystalline  bodies  as 
compared  with  colloids  of  all  kinds,  whether  living  or  dead.  But  it  so  happens 
that  the  great  majority  of  colloids  are  of  organic  origin;  and  out  of  them  our 
bodies,  and  our  food,  and  the  very  clothes  we  wear,  are  almost  wholly  made. 

A  crystal  "grows"  by  deposition  of  new  molecules,  one  by  one 
and  layer  by  layer,  each  one  superimposed  on  the  solid  substratum 

*  See  {int.  al.)  A.  Frey-Wyssling,  Subrnikroskopische  Morphologie  des  Protoplasmas, 
Berlin,  1938;  cf.  Nature,  June  10,  1939,  p.  965;  also  A.  R.  Moore,  in  Scientia, 
LXii,  July  1,  1937.  On  the  nature  of  viscid  fluid  threads,  cf.  Larmor,  Nature, 
July  11,  1936,  p.  74. 

f  Micella,  or  micula,  diminutive  of  mica,  a  crumb,  grain  or  morsel — ynica  panis, 
salis,  turis,  etc.  Nageli  used  the  word  to  mean  an  aggregation  of  molecules,  as 
the  molecule  is  an  aggregation  of  atoms;  the  one,  however,  is  a  physical  and  the 
other  a  chemical  concept.  Roughly  speaking,  we  may  think  of  micellae  as  varying 
from  about  1  to  200 /x/x;  they  play  a  corresponding  part  in  the  "disperse  phase" 
of  a  colloid  to  that  played  by  the  molecules  in  an  ordinary  solution.  The  macro- 
molecules  of  modern  chemistry  are  sometimes  distinguished  from  these  as  still 
larger  aggregates.  See  Carl  Nageli,  Das  Mikroskop  (2nd  ed.),  1877;  Theorie  der 
Gahrung,  1879. 


348  THE  FORMS  OF  CELLS  [ch. 

already  formed.  Each  particle  would  seem  to  be  influenced  only 
by  the  particles  in  its  immediate  neighbourhood,  and  to  be  in 
a  state  of  freedom  and  independence  from  the  influence,  either 
direct  or  indirect,  of  its  remoter  neighbours.  So  Lavoisier  was 
the  first  to  say.  And  as  Kelvin  and  others  later  on  explained 
the  formation  and  the  resulting  forms  of  crystals,  so  wo  beheve 
that  each  added  particle  takes  up  its  position  in  relation  to  its 
immediate  neighbours  already  arranged,  in  the  holes  and  corners 
that  their  arrangement  leaves,  and  in  closest  contact  with  the 
greatest  number*;  hence  we  may  repeat  or  imitate  this  process  of 
arrangement,  with  great  or  apparently  even  with  precise  accuracy 
(in  the  case  of  the  simpler  crystalhne  systems),  by  piling  up  spherical 
pills  or  grains  of  shot.  In  so  doing,  we  must  have  regard  to  the 
fact  that  each  particle  must  drop  into  the  place  where  it  can  go 
most  easily,  or  where  no  easier  place  offers.  In  more  technical 
language,  each  particle  is  free  to  take  up,  and  does  take  up,  its 
position  of  least  potential  energy  relative  to  those  already  there: 
in  other  words,  for  each  particle  motion  is  induced  until  the  energy 
of  the  system  is  so  distributed  that  no  tendency  or  resultant  force 
remains  to  move  it  more.  This  has  been  shewn  to  lead  to  the 
production  of  plane  surfaces  f  (in  all  cases  where,  by  the  hmitation 
of  material,  surfaces  must  occur);  where  we  have  planes,  there 
straight  edges  and  solid  angles  must  obviously  occur  also,  and,  if 
equihbrium  is  to  follow,  must  occur  symmetrically.  Our  pihng  up 
of  shot  to  make  mimic  crystals  gives  us  visible  demonstration  that 
the  result  is  actually  to  obtain,  as  in  the  natural  crystal,  plane 
surfaces  and  sharp  angles  symmetrically  disposed. 

*  Cf.  Kelvin,  On  the  molecular  tactics  of  a  crystal,  The  Boyle  Lecture,  Oxford, 
1893;  Baltimore  Lectures,  1904,  pp.  612-642.  Here  Kelvin  was  mainly  following 
Bravais's  (and  Frankenheim's)  theory  of  "space-lattices,"  but  he  had  been  largely 
anticipated  by  the  crystallographers.  For  an  account  of  the  development  of  the 
subject  in  modern  crystallography,  by  Sohncke,  von  Fedorow,  8chonfliess,  Barlow 
and  others,  see  (e.g.)  Tutton's  Crystallography,  and  the  many  papers  by  W.  E.  Bragg 
and  others. 

t  In  a  homogeneous  crystalline  arrangement,  symmetry  compels  a  locus  of  one 
property  to  be  a  plane  or  set  of  planes ;  the  locus  in  this  case  being  that  of  least 
surface  potential  energy.  Crystals  "seem  to  be,  as  it  were,  the  Elemental  Figures, 
or  the  A  B  C  of  Nature's  workmg,  the  reason  of  whose  curious  Geometrical  Forms 
(if  I  may  so  call  them)  is  very  easily  explicable"  (Robert  Hooke,  Posthumous  Works^ 
1745,  p.  280). 


V]  OF  INTUSSUSCEPTION  349 

But  the  living  cell  grows  in  a  totally  different  way,  very  much 
as  a  piece  of  glue  swells  up  in  water,  by  "imbibition,"  or  by  inter- 
penetration  into  and  throughout  its  entire  substance.  The  semi- 
fluid colloid  mass  takes  up  water,  partly  to  combine  chemically 
with  its  individual  molecules*;  partly  by  physical  diffusion  into 
the  interstices  between  molecules  or  micellae,  and  partly,  as  it  would 
seem,  in  other  ways;  so  that  the  entire  phenomenon  is  a  complex 
and  even  an  obscure  onef.  But,  so  far  as  we  are  concerned,  the 
net  result  is  very  simple.  For  the  equilibrium,  or  tendency  to 
equilibrium,  of  fluid  pressure  in  all  parts  of  its  interior  while  the 
process  of  imbibition  is  going  on,  the  constant  rearrangement  of  its 
fluid  mass,  the  contrast  in  short  with  the  crystalhne  method  of 
growth  where  each  particle  comes  to  rest  to  move  (relatively  to  the 
whole)  no  more,  lead  the  mass  of  jelly  to  swell  up  very  much  as  a 
bladder  into  which  we  blow  air,  and  so,  by  a  graded  and  harmonious 
distribution  of  forces,  to  assume  everywhere  a  rounded  and  more 
or  less  bubble-hke  external  form  J.  So,  when  the  same  school  of 
older  naturahsts  called  attention  to  a  new  distinction  or  contrast  of 
form  between  organic  and  inorganic  objects,  in  that  the  contours 
of  the  former  tended  to  roundness  and  curvature,  and  those  of  the 
latter  to  be  bounded  by  straight  lines,  planes  and  sharp  angles,  we 
see  that  this  contrast  was  not  a  new  and  different  one,  but  only 
another  aspect  of  their  former  statement,  and  an  immediate  con- 
sequence of  the  difference  between  the  processes  of  agglutination 
and  intussusception  §. 

So  far  then  as  growth  goes  on  undisturbed  by  pressure  or  other 
external  force,  the  fluidity  of  the  protoplasm,  its  mobility  internal 

*  This  is  what  Graham  called  the  water  of  gelatination,  on  the  analogy  of  water 
of  crystallisation;   Chem.  and  Phys.  Researches,  p.  597. 

t  On  this  important  phenomenon,  see  J.  R.  Katz,  Oesetze  der  Quellung,  Dresden, 
1916.  Swelling  is  due  to  "concentrated  solution,"  and  is  accompanied  by  increase 
of  volume  and  liberation  of  energy,  as  when  the  Egyptians  split  granite  by  the 
swelling  of  wood. 

X  Here,  in  a  non- crystalline  or  random  arrangement  of  particles,  symmetry 
ensures  that  the  potential  energy  shall  be  the  same  per  unit  area  of  all  surfaces; 
and  it  follows  from  geometrical  considerations  that  the  total  surface  energy  will 
be  least  if  the  surface  be  spherical. 

§  Intussusception  has  its  shades  of  meaning;  it  is  excluded  from  the  idea  of  a 
crystalline  body,  but  not  limited  to  the  ordinary  conception  of  a  colloid  one.  When 
new  micellar  strands  become  interwoven  in  the  micro-structure  of  a  cellulose  cell- 
wall,  that  is  a  special  kind  of  "intussusception." 


350  THE  FORMS  OF  CELLS  [ch. 

and  external*,  and  the  way  in  which  particles  move  freely  hither 
and  thither  within,  all  manifestly  tend  to  the  production  of  swelhng, 
rounded  surfaces,  and  to  their  great  predominance  over  plane  sur- 
faces in  the  contours  of  the  organism.  These  rounded  contours 
will  tend  to  be  preserved  for  a  while,  in  the  case  of  naked  protoplasm 
by  its  viscosity,  and  in  presence  of  a  cell-wall  by  its  very  lack  of 
fluidity.  In  a  general  way,  the  presence  of  curved  boundary 
surfaces  will  be  especially  obvious  in  the  unicellular  organisms,  and 
generally  in  the  external  form  of  all  organisms,  and  wherever 
mutual  pressure  between  adjacent  cells,  or  other  adjacent  parts, 
has  not  come  into  play  to  flatten  the  rounded  surfaces  into  planes. 

The  swelling  of  any  object,  organic  or  inorganic,  living  or  dead,  is  bound  to 
be  influenced  by  any  lack  of  structural  symmetry  or  homogeneity  f.  We 
may  take  it  that  all  elongated  structures,  such  as  hairs,  fibres  of  silk  or  cotton, 
fibrillae  of  tendon  and  connective  tissue,  have  by  virtue  of  their  elongation 
an  invisible  as  well  as  a  visible  polarity.  Moreover,  the  ultimate  fibrils  are 
apt  to  be  invested  by  a  protein  different  from  the  "collagen"  within,  and 
liable  to  swell  more  or  to  swell  less.  In  ordinary  tendons  there  is  a  "reticular 
sheath,"  which  swells  less,  and  is  apt  to  burst  under  pressure  from  within; 
it  breaks  into  short  lengths,  and  when  the  strain  is  relieved  these  roll  back, 
and  form  the  familiar  annuli.  Another  instance  is  the  tendency  to  swell  of 
the  "macro- molecules"  of  many  polymerised  organic  bodies,  proteins  among 
them. 

But  the  rounded  contours  which  are  assumed  and  exhibited  by 
a  piece  of  hard  glue  when  we  throw  it  into  water  and  see  it  expand 
as  it  sucks  the  water  up,  are  not  near  so  regular  nor  so  beautiful  as 
are  those  which  appear  when  we  blow  a  bubble,  or  form  a  drop,  or 
even  pour  water  into  an  elastic  bag.  For  these  curving  contours 
depend  upon  the  properties  of  the  bag  itself,  of  the  film  or  membrane 
which  contains  the  mobile  gas,  or  which  contains  or  bounds  the 
mobile  liquid  mass.  And  hereby,  in  the  case  of  the  fluid  or  semifluid 
mass,  we  are  introduced  to  the  subject  of  surface-tension:  of  which 
indeed  we  have  spoken  in  the  preceding  chapter,  but  which  we  must 
now  examine  with  greater  care. 

*  The  protoplasm  of  a  sea-urchin's  egg  has  a  viscosity  only  about  four  times, 
and  that  of  various  plants  not  more  than  ten  to  twenty  times,  that  of  water  itself. 
See,  for  a  general  discussion,  L.  V.  Heilbrunn,  Colloid  Symposium  Monograph^  1928. 

t  D.  Jordan  Lloyd  and  R.  H.  Marriott,  The  swelling  of  structural  proteins, 
Proc.  U.S.  (B),  No.  810,  pp.  439-44.3,  1935. 


V]  OF  SURFACE  TENSION  351 

Among  the  forces  which  determine  the  forms  of  cells,  whether 
they  be  sohtary  or  arranged  in  contact  with  one  another,  this  force 
of  surface-tension  is  certainly  of  great,  and  is  probably  of  paramount, 
importance.  But  while  we  shall  try  to  separate  out  the  phenomena 
which  are  directly  due  to  it,  we  must  not  forget  that,  in  each 
particular  case,  the  actual  conformation  which  we  study  may  be, 
and  usually  is,  the  more  or  less  complex  resultant  of  surface-tension 
acting  together  with  gravity,  mechanical  pressure,  osmosis,  or  other 
physical  forces.  The  peculiar  beauty  of  a  soap-bubble,  solitary  or 
in  collocation,  depends  on  the  absence  (to  all  intents  and  purposes) 
of  these  ahen  forces  from  the  field ;  hence  Plateau  spoke  of  the  films 
which  were  the  subject  of  his  experiments  as  "lames  fluides  sans 
pesanteur.''  The  resulting  form  is  in  such  a  case  so  pure,  and  simple 
that  we  come  to  look  on  it  as  wellnigh  a  mathematical  abstraction. 

Surface-tension,  then,  is  that  force  by  which  we  explain  the  form 
of  a  drop  or  of  a  bubble,  of  the  surfaces  external  and  internal  of 
a  "froth"  or  collocation  of  bubbles,  and  of  many  other  things  of 
like  nature  and  in  hke  circumstances*.  It  is  a  property  of  Hquids 
(in  the  sense  at  least  with  which  our  subject  is  concerned),  and  it 
is  manifested  at  or  very  near  the  surface,  where  the  liquid  comes 
into  contact  with  another  hquid,  a  solid  or  a  gas.  We  note  here 
that  the  term  surface  is  to  be  interpreted  in  a  wide  sense;  for 
wherever  we  have  solid  particles  embedded  in  a  fluid,  wherever  we 
have  a  non-homogeneous  fluid  or  semi-fluid,  or  a  "two-phase  colloid " 
such  as  a  particle  of  protoplasm,  wherever  we  have  the  presence  of 
"impurities"  as  in  a  mass  of  molten  metal,  there  we  have  always 
to  bear  in  mind  the  existence  of  surfaces  and  of  surface-phenomena, 
not  only  on  the  exterior  of  the  mass  but  also  throughout  its  inter- 
stices, wherever  like  and  unlike  meet. 

*  The  idea  of  a  "surface-tension"  in  liquids  was  first  enunciated  by  Segner,  and 
ascribed  by  him  to  forces  of  attraction  whose  range  of  action  was  so  small  "ut 
nullo  adhuc  sensu  percipi  potuerat"  {Defiguris  super ficier um  fiuidarum,  in  Comment. 
Soc.  Roy.  GoUingen,  1751,  p.  301).  Hooke,  in  the  Micrographia  (1665,  Obs.  viii, 
etc.),  had  called  attention  to  the  globular  or  spherical  form  of  the  little  morsels 
of  steel  struck  off  by  a  flint,  and  had  shewn  how  to  make  a  powder  of  such  spherical 
grains,  by  heating  fine  filings  to  melting  point,  "This  Phaenomenon"  he  said 
"proceeds  from  a  propriety  which  belongs  to  all  kinds  of  fluid  Bodies  more  or  less, 
and  is  caused  by  the  Incongruity  of  the  Atnbient  and  included  Fluid,  which  so 
acts  and  modulates  each  other,  that  they  acquire,  as  neer  as  is  possible,  a  spherical 
or  globular  form " 


352  THE  FORMS  OF  CELLS  [ch. 

A  liquid  in  the  mass  is  devoid  of  structure ;  it  is  homogeneous,  and 
without  direction  or  polarity.  But  the  very  concept  of  surface- 
tension  forbids  this  to  be  true  of  the  surface-layer  of  a  body  of  liquid, 
or  of  the  "interphase"  between  two  liquids,  or  of  any  film,  bubble, 
drop,  or  capillary  jet  or  stream.  In  all  these  cases,  and  more  empha- 
tically in  the  case  of  a  "monolayer,"  even  the  liquid  has  a  structure 
of  its  own ;  and  we  are  reminded  once  again  of  how  largely  the  living 
organism,  whether  high  or  low,  is  composed  of  colloid  matter  in 
precisely  such  forms  and  structural  conditions. 

Surface-tension  is  due  to  molecular  force  * :  to  force,  that  is  to  say, 
arising  from  the  action  of  one  molecule  upon  another;  and  since 
we  can  only  ascribe  a  small  "sphere  of  action"  to  each  several 
molecule,  this  force  is  manifested  only  within  a  narrow  range. 
Within  the  interior  of  the  liquid  mass  we  imagine  that  such  molecular 
interactions  negative  one  another ;  but  at  and  near  the  free  surface, 
within  a  layer  or  film  approximately  equal  to  the  range  of  the 
molecular  force — or  to  the  radius  of  the  aforesaid  "  sphere  of  action  " 
— there  is  a  lack  of  equilibrium  and  a  consequent  manifestation  of 
force. 

The  action  of  the  molecular  forces  has  been  variously  explained. 
But  one  simple  explanation  (or  mode  of  statement)  is  that  the 
molecules  of  the  surface-layer  are  being  constantly  attracted  into 
the  interior  by  such  as  are  just  a  little  more  deeply  situated;  the 
surface  shrinks  as  molecules  keep  quitting  it  for  the  interior,  and 
this  surface-shrinkage  exhibits  itself  as  a  surface-tension.  The  process 
continues  till  it  can  go  no  farther,  that  is  to  say  until  the  surface 
itself  becomes  a  "minimal  areaf."  This  is  a  sufficient  description 
of  the  phenomenon  in  cases  where  a  portion  of  liquid  is  subject  to 
no  other  than  its  own  molecular  forces,  and  (since  the  sphere  has, 

*  While  we  explain  certain  phenomena  of  the  organism  by  reference  to  atomic 
or  molecular  forces,  the  following  words  of  Du  Bois  Reymond's  seem  worth 
recalling:  ' Naturerkennen  ist  Zuriickfiihren  der  Veranderungen  in  der  Korperwelt 
auf  Bewegung  von  Atomen,  die  durch  deren  von  der  Zeit  unabhangige  Centralkrafte 
bewirkt  werden,  oder  Auflosung  der  Naturkrafte  in  Mechanik  der  Atome.  Es  ist 
eine  psychologische  Erfahrungstatsache  dass,  wo  solche  Auflosung  gelangt,  unser 
Causalbediirfniss  vorlaiifig  sioh  befriedigt  fiihlt"  (Ueber  die  Grenzen  des  Natnr- 
erkennens,  Leipzig,  1873). 

t  There  must  obviously  be  a  certain  kinetic  energy  in  the  molecules  within 
the  drop,  to  balance  the  forces  which  are  trying  to  contract  and  diminish  the 
surface. 


V]  OF  SURFACE  ENERGY  353 

of  all  solids,  the  least  surface  for  a  given  volume)  it  accounts  for 
the  spherical  form  of  the  raindrop*,  of  the  grain  of  shot,  or  of  the 
living  cell  in  innumerable  simple  organisms  f.  It  accounts  also,  as 
we  shall  presently  see,  for  many  much  more  complicated  forms, 
manifested  under  less  simple  conditions. 

Let  us  note  in  passing  that  surface-tension  is  a  comparatively 
small  force  and  is  easily  measurable:  for  instance  that  between 
water  and  air  is  equivalent  to  but  a  few  grains  per  linear  inch,  or 
a  few  grammes  per  metre.  But  this  small  tension,  when  it  exists 
in  a  curved  surface  of  great  curvature,  such  as  that  of  a  minute  drop, 
gives  nse  to  a  very  great  pressure,  directed  inwards  towards  the 
centre  of  curvature.  We  may  easily  calculate  this  pressure,  and  so 
satisfy  ourselves  that,  when  the  radius  of  curvature  approaches 
molecular  dimensions,  the  pressure  is  of  the  order  of  thousands  of 
atmospheres — a  conclusion  which  is  supported  by  other  physical 
considerations. 

The  contraction  of  a  liquid  surface,  and  the  other  phenomena  of 
surface-tension,  involve  the  doing  of  work,  and  the  power  to  do 
work  is  what  we  call  Energy.  The  whole  energy  of  the  system  is 
diffused  throughout  its  molecules,  as  is  obvious  in  such  a  simple 
case  as  we  have  just  considered;  but  of  the  whole  stock  of  energy 
only  the  part  residing  at  or  very  near  the  surface  normally  manifests 
itself  in  work,    and  hence  we  speak  (though  the  term  be  open  to 

*  Raindrops  must  be  spherical,  or  they  would  not  produce  a  rainbow;  and  the 
fact  that  the  upper  part  of  the  bow  is  the  brightest  and  sharpest  shews  that  the 
higher  raindrops  are  more  truly  spherical,  as  well  as  smaller  than  the  lower  ones. 
So  also  the  smallest  dewdrops  are  found  to  be  more  iridescent  than  the  large,  shewing 
that  they  also  are  the  more  truly  spherical;  cf.  T.  W.  Backhouse,  in  Monthly 
Meteorol.  Mag.  March,  1879.  Mercury  has  a  high  surface-tension,  and  its  globules 
are  very  nearly  round. 

t  That  the  offspring  of  a  spherical  cell  (whether  it  be  raindrop,  plant  or  animal) 
should  be  also  a  spherical  cell,  would  seem  to  need  no  other  explanation  than  that 
both  are  of  identical  substance,  and  each  subject  to  a  similar  equilibrium  of 
surface-forces;  but  the  biologists  have  been  apt  to  look  for  a  subtler  reason. 
Giglio-Tos,  speaking  of  a  sea-urchin's  dividing  egg,  asks  why  the  daughter-cells 
are  spherical  like  the  mother-cell,  and  finds  the  reason  in  "heredity":  "Wenn  also 
die  letztere  (d.  i.  die  Mutterzelle)  eine  spharische  Form  besass,  so  nehmen  auch  die 
Tochterzellen  dieselbe  ein;  ware  urspriinglich  eine  kubische  Form  vorhanden, 
so  wurden  also  auch  die  Tochterzellen  dieselbe  auch  aneignen.  Die  Ursache  warum 
die  Tochterzellen  die  spharische  Form  anzunehmen  trachten  liegt  darin,  dasa 
diese  die  Ur-  und  Grundform  alter  Zellen  ist,  sowohl  bet  Tieren  ivie  bei  den  Pflanzen'' 
[Arch.f.  Entw.  Mech.  Li,  p.  115,  1922). 


354  THE  FORMS  OF  CELLS  [ch. 

some  objection)  of  a  specific  surface-energy.  Surface-energy,  and 
the  way  it  is  increased  and  multiplied  by  the  multiphcation  of 
surfaces  due  to  the  subdivision  of  the  tissues  into  cells,  is  of  the 
highest  interest  tq  the  physiologist;  and  even  the  morphologist 
cannot  pass  it  by.  For  the  one  finds  surface-energy  present,  often 
perhaps  paramount,  in  every  cell  of  the  body ;  and  the  other  may  find, 
if  he  will  only  look  for  it,  the  form  of  every  solitary  cell,  hke  that  of 
any  other  drop  or  bubble,  related  to  if  not  controlled  by  capillarity. 
The  theory  of  "capillarity,"  or  "surface-energy,"  has  been  set  forth 
with  the  utmost  possible  lucidity  by  Tait  and  by  Clerk  Maxwell,  on 
whom  the  following  paragraphs  are  based:  they  having  based  their 
teaching  on  that  of  Gauss*,  who  rested  on  Laplace. 

Let  E  be  the  whole  potential  energy  of  a  mass  M  of  hquid;  let 
Cq  be  the  energy  per  unit  mass  of  the  interior  hquid  (we  may  call  it 
the  internal  energy);  and  let  e  be  the  energy  per  unit  mass  for  a 
layer  of  the  skin,  of  surface  S,  of  thickness  t,  and  density  p  (e  being 
what  we  call  the  surface-energy).  It  is  obvious  that  the  total  energy 
consists  of  the  internal  plus  the  surface-energy,  and  that  the  former 
is  distributed  through  the  whole  mass,  minus  its  surface  layers. 
That  is  to  say,  in  mathematical  language, 

E={M  -S.  l.tp)  e^  +  S  .  I^tpe. 

But  this  is  equivalent  to  writing : 

=  MeQ  +  S  .I.tp{e-  eo); 

and  this  is  as  much  as  to  say  that  the  total  energy  of  the  system 
may  be  taken  to  consist  of  two  portions,  one  uniform  throughout 
the  whole  mass,  and  another,  which  is  proportional  on  the  one  hand 
to  the  amount  of  surface,  and  on  the  other  hand  to  the  difference 
between  e  and  Cq,  that  is  to  say  to  the  difference  between  the  unit 
values  of  the  internal  and  the  surface  energy. 

It  was  Gauss  who  first  shewed  how,  from  the  mutual  attractions 
between  all  the  particles,  we  are  led  to  an  expression  for  what  we 

*  See  Gauss's  Principia  generalia  Theoriae  Figurae  Fluidorum  in  statu  equilibriif 
Gottingen,  1830.  The  historical  student  will  not  overlook  the  claims  to  priority 
of  Thomas  Young,  in  his  Essay  on  the  cohesion  of  fluids,  Phil.  Trans.  1805;  see 
the  account  given  in  his  Life  by  Dean  Peacock,  18.55,  pp.  199-210. 


V]  OF  SURFACE  ENERGY  355 

no\v'  call  the  potential  energy*  of  the  system;  and  we  know,  as  a 
fundamental  theorem  of  dynamics,  as  well  as  of  molecular  physics, 
that  the  potential  energy  of  the  system  tends  to  a  minimum,  and 
finds  in  that  minimum  its  stable  equihbrium. 

We  see  in  our  last  equation  that  the  term  Mcq  is  irreducible,  save 
by  a  reduction  of  the  mass  itself.  But  the  other  term  may  be 
diminished  (1)  by  a  reduction  in  the  area  of  surface,  S,  or  (2)  by 
a  tendency  towards  equahty  of  e  and  ^q  ,  that  is  to  say  by  a  diminu- 
tion of  the  specific  surface  energy,  e. 

These  then  are  the  two  methods  by  which  the  energy  of  the 
system  will  manifest  itself  in  work.  The  one,  which  is  much  the 
more  important  for  our  purposes,  leads  always  to  a  diminution  of 
surface,  to  the  so-called  "principle  of  minimal  areas";  the  other, 
which  leads  to  the  lowering  (under  certain  circumstances)  of  surface 
tension,  is  the  basis  of  the  theory  of  Adsorption,  to  which  we  shall 
have  some  occasion  to  refer  as  the  modus  operandi  in  the  develop- 
ment of  a  celi-wall,  and  in  a  variety  of  other  histological  phenomena. 
In  the  technical  phraseology  of  the  day,  the  '^capacity  factor"  is 
involved  in  the  one  case,  and  the  "intensity  factor"  in  the  otherf. 

Inasmuch  as  we  are  concerned  with  the  form  of  the  cell,  it  is  the 
former  which  becomes  our  main  postulate:  telhng  us  that  the 
energy-equations  of  the  surface  of  a  cell,  or  of  the  free  surfaces  of 
cells  in  partial  contact,  or  of  the  partition-surfaces  of  cells  in  contact 
with  one  another,  all  indicate  a  minimum  of  potential  energy  in  the 
system,  by  which  minimal  condition  the  system  is  brought,  ipso 
facto,  into  equihbrium.  And  we  shall  not  fail  to  observe,  with 
something  more  than  mere  historical  interest  and  curiosity,  how 


*  The  word  Energy  was  substituted  for  the  old  vis  viva  by  Thomas  Young  early 
in  the  nineteenth  century,  and  was  used  by  James  Thomson,  Lord  Kelvin's  brother, 
about  1852,  to  mean,  more  generally,  "capacity  for  doing  work."  The  term  potential, 
or  latent,  in  contrast  to  actual  energy,  in  other  words  the  distinction  between  "  energy 
of  activity  and  energy  of  configuration,"  was  proposed  by  Macquorn  Rankine,  and 
suggested  to  him  by  Aristotle's  use  of  8vva/j.L$  and  evepyeia;  see  Rankine's  paper 
On  the  general  law  of  the  transformation  of  energy,  Phil.  Soc.  Glasgow,  Jan. 
5,  1853,  cf.  ibid.  Jan.  23,  1867,  and  Phil.  Mag.  (4),  xxvii,  p.  404,  1864.  The  phrase 
potential  energy  was  at  once  adopted,  but  kinetic  was  substituted  for  actical  by 
Thomson  and  Tait. 

t  The  capacity  factor,  inasmuch  as  it  leads  to  diminution  of  surface,  is  responsible 
for  the  concrescence  of  droplets  into  drops,  of  microcrystals  into  larger  units,  for 
the  flocculation  of  colloids,  and  for  many  other  similar  "changes  of  state." 


356  THE  FORMS  OF  CELLS  [ch. 

deeply  and  intrinsically  there  enter  into  this  whole  class  of  problems 
the  method  of  maxima  and  minima  discovered  by  Fermat,  the  "loi 
universelle  de  repos"  of  Maupertuis,  "dont  tous  les  cas  d'equilibre 
dans  la  statique  ordinaire  ne  sont  que  des  cas  particuhers",  and  the 
lineae  curvae  maximi  miniynive  proprietatibus  gaudentes  of  Euler,  by 
which  principles  these  old  natural  philosophers  explained  correctly 
a  multitude  of  phenomena,  and  drew  the  lines  whereon  the  founda- 
tions of  great  part  of  modern  physics  are  well  and  truly  laid.  For 
that  physical  laws  deal  with  minima  is  very  generally  true,  and  is 
highly  characteristic  of  them.  The  hanging  chain  so  hangs  that  the 
height  of  its  centre  of  gravity  is  a  minimum ;  a  ray  of  hght  takes 
the  path,  however  devious,  by  which  the  time  of  its  journey  is  a 
minimum ;  two  chemical  substances  in  reaction  so  behave  that  their 
thermodynamic  potential  tends  to  a  minimum,  and  so  on.  The 
natural  philosophers  of  the  eighteenth  century  were  engrossed  in 
minimal  problems ;  and  the  differential  equations  which  solve  them 
nowadays  are  among  the  most  useful  and  most  characteristic  equa- 
tions in  mathematical  physics. 

"Voici,"  said  Maupertuis,  "dans  un  assez  petit  volume  a  quoi  je 
reduis  mes  ouvrages  mathematiques ! "  And  when  Lagrange,  fol- 
lowing Euler 's  lead*,  conceived  the  principle  of  least  action,  he 
regarded  it  not  as  a  metaphysical  principle  but  as  "un  resultat 
simple  et  general  des  lois  de  la  mecaniquef."  The  principle  of 
least  action  J  explains  nothing,  it  tells  us  nothing  of  causation, 
yet  it  illuminates  a  host  of  things.  Like  Maxwell's  equations  and 
other  such  flashes  of  genius  it  clarifies  our  knowledge,  adds  weight 
to  our  observations,  brings  order  into  our  stock-in-trade  of  facts. 
It  embodies  and  extends  that  "law  of  simplicity"  which  Borelli 
was  the  first  to  lay  down:  "Lex  perpetua  Naturae  est  ut  agat 
minimo  labore,  mediis  et  modis  simplicissimis,  facillimis,  certis  et 

*  Euler,  Traite  des  Isoperimetres,  Lausanne,  1744. 

t  Lagrange,  Mecanique  Analytique  (2),  ii,  p.  188;   ed.  in  4to,  1788. 

{  This  profound  conception,  not  less  metaphysical  in  the  outset  than  physical, 
began  in  the  seventeenth  century  with  Fermat,  who  shewed  (in  1629)  that  a  ray 
of  light  followed  the  quickest  path  available,  or,  as  Leibniz  put  it,  via  omnium 
facillima;  it  was  over  this  principle  that  Voltaire  quarrelled  with  Euler  and 
Maupertuis.  The  mathematician  will  think  also  of  Hamilton's  restatement  of  the 
principle,  and  of  its  extension  to  the  theory  of  probabilities  by  Boltzmann  and 
Willard  Gibbs.  Cf.  (int.  al.)  A.  Mayer,  Geschichte  des  Prinzips  der  kleinsten  Action, 
1877. 


vj  THE  MEANING  OF  SYMMETRY  357 

tutis:  evitando,  quam  maxime  fieri  potest,  incommoditates  et 
prolixitates."  The  principle  of  least  action  grew  up,  and  grew 
quickly,  out  of  cruder,  narrower  notions  of  "least  time"  or  "least 
space  or  distance."  Nowadays  it  is  developing  into  a  principle  of 
"least  action  in  space-time,"  which  shall  still  govern  and  predict 
the  motions  of  the  universe.  The  infinite  perfection  of  Nature  is 
expressed  and  reflected  in  these  concepts,  and  Aristotle's  great 
aphorism  that  "Nature  does  nothing  in  vain"  lies  at  the  bottom 
of  them  all. 

In  all  cases  where  the  principle  of  maxima  and  minima  comes 
into  play,  as  it  conspicuously  does  in  films  at  rest  under  surface- 
tension,  the  configurations  so  produced  are  characterised  by  obvious 
and  remarkable  symmetry'^.  Such  symmetry  is  highly  characteristic 
of  organic  forms,  and  is  rarely  absent  in  living  things — save  in  such 
few  cases  as  Amoeba,  where  the  rest  and  equilibrium  on  which 
symmetry  depends  are  likewise  lacking.  And  if  we  ask  what 
physical  equilibrium  has  to  do  with  formal  symmetry  and  structural 
regularity,  the  reason  is  not  far  to  seek,  nor  can  it  be  better  put 
than  in  these  words  of  Mach'sf:  "In  every  symmetrical  system 
every  deformation  that  tends  to  destroy  the  symmetry  is  com- 
plemented by  an  equal  and  opposite  deformation  that  tends  to 
restore  it.  In  each  deformation,  positive  and  negative  work  is  done. 
One  condition,  therefore,  though  not  an  absolutely  sufficient  one, 
that  a  niaximum  or  minimum  of  work  corresponds  to  the  form  of 
equiUbrium,  is  thus  supj)ned  by  symmetry.  Regularity  is  successive 
symmetry;  there  is  no  reason,  therefore,  to  be  astonished  that  the 
forms  of  equiUbrium  are  often  symmetrical  and  regular." 

A  crystal  is  the  perfection  of  symmetry  and  of  regularity; 
symmetry  is  displayed  in  its  external  form,  and  regularity  revealed 
in  its  internal  lattices.  Complex  and  obscure  as  the  attractions, 
rotations,  vibrations  and  what  not  within  the  crystal  may  be,  we 
rest  assured  that  the  configuration,  repeated  again  and  again,  of 

*  On  the  mathematical  side,  cf.  Jacob  Nteiner,  Einfache  Beweise  der  isoperi- 
metrischen  Hauptsatze,  Abh.  k.  Akad.  Wisa.  Berlin,  xxiii,  pp.  116-135,  1836  (1838). 
On  the  biological  side,  see  {int.  al.)  F.  M.  Jaeger,  Lectures  on  the  Principle  of  Symmetry, 
and  its  application  to  the  natural  sciences,  Amsterdam,  1917;  also  F.  T.  Lewis, 
Symmetry. .  .in  evolution,  Amer.  Nat.  lvii,  pp.  5-41,  1923. 

f  Science  of  Mechanics,  1902,  p.  395;  see  also  Mach's  article  Ueber  die  physika- 
lische  Bedeutung  der  Gesetze  der  Symmetrie,  Lotos,  xxi,  pp.  139-147,  1871. 


358  THE  FORMS  OF  CELLS  [ch. 

the  component  atoms  is  precisely  that  for  which  the  energy  is  a 
minimum;  and  we  recognise  that  this  minimal  distril^ution  is  of 
itself  tantamount  to  symmetry  and  to  stability. 

Moreover,  the  principle  of  least  action  is  but  a  setting  of  a  still 
more  universal  law — that  the  world  and  all  the  parts  thereof  tend 
ever  to  pass  from  less  to  more  probable  configurations;  in  which 
the  physicist  recognises  the  principle  of  Clausius,  or  second  law  of 
thermodynamics,  and  with  which  the  biologist  must  somehow 
reconcile  the  whole  "theory  of  evolution." 

As  we  proceed  in  our  enquiry,  and  especially  when  we  approach 
the  subject  of  tissues,  or  agglomerations  of  cells,  we  shall  have  from 
time  to  time  to  call  in  the  help  of  elementary  mathematics.  But 
already,  with  very  Httle  mathematical  help,  we  find  ourselves  in  a 
position  to  deal  with  some  simple  examples  of  organic  forms. 

When  we  melt  a  stick  of  sealing-wax  in  the  flame,  surface-tension 
(which  was  ineffectively  present  in  the  solid  but  finds  play  in  the 
now  fluid  mass)  rounds  off  its  sharp  edges  into  curves,  so  striving 
tbwards  a  surface  of  minimal  area ;  and  in  like  manner,  by  merely 
melting  the  tip  of  a  thin  rod  of  glass,  Hooke  made  the  little  spherical 
beads  which  served  him  for  a  microscope*.  When  any  drop  of 
protoplasm,  either  over  all  its  surface  or  at  some  free  end,  as  at  the 
extremity  of  the  pseudopodium  of  an  amoeba,  is  seen  hkewise  to 
"round  itself  off,"  that  is  not  an  effect  of  "vital  contractility,"  but, 
as  Hofmeister  shewed  so  long  ago  as  1867,  a  simple  consequence  of 
surface-tension;  and  almost  immediately  afterwards  Engelmannf 
argued  on  the  same  lines,  that  the  forces  which  cause  the  contraction 
of  protoplasm  in  general  may  "be  just  the  same  as  those  which  tend 
to  make  every  non-spherical  drop  of  fluid  become  spherical."  We 
are  not  concerned  here  with  the  many  theories  and  speculations 
which  would  connect  the  phenomena  of  surface-tension  with  con- 
tractility, muscular  movement,  or  other  special  'physiological  func- 

*  Similarly,  Sir  David  Brewster  and  others  made  powerful  lenses  by  simply 
dropping  small  drops  of  Canada  balsam,  castor  oil,  or  other  strongly  refractive 
liquids,  on  to  a  glass  plate:  On  New  Philosophical  Instruments  (Description  of  a 
new  fluid  microscope),  Edinburgh,  1813,  p.  413.  See  also  Hooke's  Micrographia, 
1665;  and  Adam's  Essay  on  the  Microscope,  1798,  p.  8:  "No  person  has  carried 
the  use  of  these  globules  so  far  as  Father  Torre  of  Naples,  etc."  Leeuwenhoek. 
on  the  other  hand,  ground  his  lenses  with  exquisite  skill. 

t  Beitrage  zur  Physiologie  des  Protoplasma,  Pfluger's  Archlv,  ii,  p.  307,  1869. 


V]  OF  SURFACE  ACTION  359 

tions,  but  we  find  ample  room  to  trace  the  operation  of  the  same 
cause  in  producing,  under  conditions  of  rest  and  equiUbrium,  certain 
definite  and  inevitable  forms. 

It  is  of  great  importance  to  observe  that  the  living  cell  is  one 
of  those  cases  where  the  phenomena  of  surface-tension  are  by  no 
means  limited  to  the  outer  surface;  for  within  the  heterogeneous 
emulsion  of  the  cell,  between  the  protoplasm  and  its  nuclear  and 
other  contents,  and  in  the  "alveolar  network"  of  the  cytoplasm 
itself  (so  far  as  that  alveolar  structure  is  actually  present  in  life), 
we  have  a  multitude  of  interior  surfaces;  and,  especially  among 
plants,  we  may  have  large  internal  "interfacial  contacts"  between 
the  protoplasm  and  its  included  granules,  or  its  vacuoles  filled  with 
the  "cell-sap."  Here  we  have  a  great  field  for  surface-action;  and 
so  long  ago  as  1865,  Nageli  and  Schwendener  shewed  that  the 
streaming  currents  of  plant  cells  might  be  plausibly  explained  by 
this  phenomenon.  Even  ten  years  earlier,  Weber  had  remarked 
upon  the  resemblance  between  the  protoplasmic  streamings  and 
the  currents  to  be  observed  in  certain  inanimate  drops  for  which 
no  cause  but  capillarity  could  be  assigned*.  What  sort  of  chemical 
changes  lead  up  to,  or  go  hand  in  hand  with,  the  variations  of 
surface-tension  in  a  hving  cell,  is  a  vastly  important  question.  It 
is  hardly  one  for  us  to  deal  with ;  but  this  at  least  is  clear,  that  the 
phenomenon  is  more  complicated  than  its  first  investigators,  such 
as  BUtschli  and  Quincke,  ever  took  it  to  be.  For  the  lowered 
surface-tension  which  leads,  say,  to  the  throwing  out  of  a  pseudo- 
podium,  is  accompanied  first  by  local  acidity,  then  by  local 
adsorption  of  proteins,  lastly  and  consequently  by  gelation;  and 
this  last  is  tantamount  to  the  formation  of  "ectoplasm" — a  step 
in  the  direction  of  encystmentf. 

The  elementary  case  of  Amoeba  is  none  the  less  a  complicated  one. 
The  "amoeboid"  form  is  the  very  negation  of  rest  or  of  equihbrium; 

*  Poggendorff's  Annalen,  xciv,  pp.  447-459,  1855.  Cf.  Strethill  Wright,  Phil. 
Mag.  Feb.  1860;   Journ.Anat.  and  Physiol,  i,  p.  337,  1867. 

t  Cf.  C,  J.  Pantin,  Journ.  Mar.  Biol.  Assoc,  xiii,  p.  24,  1923;  Journ.  Exp.  Biol. 
1923  and  1926;  S.  0.  Mast,  Jo^lrn.  Morph.  xli,  p.  347,  1926;  and  0.  W.  Tiegs, 
Surface  tension  and  the  theory  of  protoplasmic  movement,  Protoplasma,  iv, 
pp.  88-139,  1928.  See  also  (int.  al.)  N.  K.  Adam,  Physics  and  Chemistry  of  Surfaces, 
1930;  also  Discussion  on  colloid  science  applied  to  biology  (passim),  Trans.  Faraday 
Soc.  XXVI,  pp.  663  seq.,  1930. 


360  THE  F0RM8  OF  CELLS  [ch. 

the  creature  is  always  moving,  from  one  protean  configuration  to 
another;  its  surface-tension  is  never  constant,  but  continually 
varies  from  here  to  there.  Where  the  surface  tension  ^is  greater, 
that  portion  of  the  surface  will  contract  into  spherical  or  spheroidal 
forms;  where  it  is  less,  the  surface  will  correspondingly  extend. 
While  generally  speaking  the  surface-energy  has  a  minimal  value, 
it  is  not  necessarily  constant.  It  may  be  diminished  by  a  rise  of 
temperature;  it  may  be  altered  by  contact  with  adjacent  sub- 
stances*, by  the  transport  of  constituent  materials  from  the  interior 
to  the  surface,  or  again  by  actual  chemical  and  fermentative  change ; 
for  within  the  cell,  the  surface-energies  developed  about  its  hetero- 
geneous contents  will  continually  vary  as  these  contents  are  affected 
by  chemical  metabolism.  As  the  colloid  materials  are  broken  down 
and  as  the  particles  in  suspension  are  diminished  in  size  the  "free 
surface-energy"  will  be  increased,  but  the  osmotic  energy  will  be 
diminished!.  Thus  arise  the  various  fluctuations  of  surface-tension, 
and  the  various  phenomena  of  amoeboid  form  and  motion,  which 
Biitschli  and  others  have  reproduced  or  imitated  by  means  of  the 
fine  emulsions  which  constitute  their  "artificial  amoebae." 

A  multitude  of  experiments  shew  how  extraordinarily  dehcate  is 
the  adjustment  of  the  surface-tension  forces,  and  how  sensitive  they 
are  to  the  least  change  of  temperature  or  chemical  state.     Thus, 

*  Haycraft  and  Carlier  pointed  out  long  ago  {Proc.  R.S.E.  xv,  pp.  220-224, 
1888)  that  the  amoeboid  movements  of  a  white  blood-corpuscle  are  only  manifested 
when  the  corpuscle  is  in  contact  with  some  solid  substance:  while  floating  freely 
in  the  plasma  or  serum  of  the  blood,  these  corpuscles  are  spherical,  that  is  to  say 
they  are  at  rest  and  in  equilibrium.  The  same  fact  was  recorded  anew  by 
Ledingham  (On  phagocytosis  from  an  adsorptive  point  of  view,  Journ.  Hygiene, 
XII,  p.  324,  1912).  On  the  emission  of  pseudopodia  as  brought  about  by  changes 
in  surface  tension,  see  also  {int.  al.)  J.  A.  Ryder,  Dynamics  in  Evolution,  1894; 
Jensen,  L'eber  den  Geotropismus  niederer  Orgahismen,  Pfliiger's  Archiv,  liii,  1893. 
Jensen  remarks  that  in  Orbitolites,  the  pseudopodia  issuing  through  the  pores  of 
the  shell  first  float  freely,  then  as  they  grow  longer  bend  over  till  they  touch  the 
ground,  whereupon  they  begin  to  display  amoeboid  and  streaming  motions. 
\'erworn  indicates  {Ally.  Physiol.  189o,  p.  429),  and  Davenport  says  {Exper. 
Morphology,  ii,  p.  376),  that  "this  persistent  clinging  to  the  substratum  is  a 
'  thigmotropic '  reaction,  and  one  which  belongs  clearly  to  the  category  of '  response '. " 
Cf.  Putter,  Thigmotaxis  bei  Protisten,  Arch.  f.  Physiol.  1900,  Suppl.  p.  247;  but 
it  is  not  clear  to  my  mind  that  to  account  for  this  simple  phenomenon  we  need 
invoke  other  factors  than  gravity  and  surface-action. 

t  Cf.  Pauli,  Allgemeine  physikalische  Chemie  d.  Zellen  u.  Gewebe,  in  Asher-Spiro's 
Ergebnisse  der  Physiologic,  1912;    Przibram,  Vitalitdt,  1913,  p.  6. 


V]  THE  FORM  OF  AMOEBA  361 

on  a  plate  which  we  have  warmed  at  one  side  a  drop  of  alcohol 
runs  towards  the  warm  area,  a  drop  of  oil  away  from  it;  and  a 
drop  of  water  on  the  glass  plate  exhibits  lively  movements  when 
we  bring  into  its  neighbourhood  a  heated  wire,  or  a  glass  rod  dipped 
in  ether*.  The  water-colour  painter  makes  good  use  of  the  surface- 
tension  effect  of  the  minutest  trace  of  ox-gall.  When  a  plasmodium 
of  Aethalium  creeps  towards  a  damp  spot  or  a  warm  spot,  or 
towards  substances  which  happen  to  be  nutritious,  and  creeps 
away  from  solutions  of  sugar  or  of  salt,  we  are  dealing  with  pheno- 
mena too  often  ascribed  to  'purposeful'  action  or  adaptation,  but 
every  one  of  which  can  be  paralleled  by  ordinary  phenomena  of 
surface-tensiont-  The  soap-bubble  itself  is  never  in  equilibrium: 
for  the  simple  reason  that  its  film,  Hke  the  protoplasm  of  Amoeba 
or  Aethalium,  is  exceedingly  heterogeneous.  Its  surface-energies 
vary  from  point  to  point,  and  chemical  changes  and  changes  of 
temperature  increase  and  magnify  the  variation.  The  surface  of 
the  bubble  is  in  continual  movement,  as  more  concentrated  portions 
of  the  soapy  fluid  make  their  way  outwards  from  the  deeper  layers ; 
it  thins  and  it  thickens,  its  colours  change,  currents  are  set  up  in 
it  and  little  bubbles  glide  over  it;  it  continues  in  this  state  of 
restless  movement  as  its  parts  strive  one  with  another  in  their 
interactioAs  towards  unattainable  equilibrium  J.  On  reaching  a 
certain  tenuity  the  bubble  bursts:  as  is  bound  to  happen  when 
the  attenuated  film  has  no  longer  the  properties  of  matter  in  mass. 

*  80  Bernstein  shewed  that  a  drop  of  mercury  in  nitric  acid  moves  towards,  or 
is  "attracted  by,"  a  crystal  of  potassium  bichromate;  Pfliiger's  Archiv,  lxxx, 
p.  628,  1900.      "^ 

t  The  surface-tension  theory  of  protoplasmic  movement  has  been  denied  by 
many.  Cf.  (e.g.)  H.  S.  Jennings,  Contributions  to  the  behaviour  of  the  lower 
organisms,  Carnegie  Instit.  1904,  pp.  130-230;  O.  P.  Dellinger,  Locomotion  of 
Amoebae,  etc.,  Journ.  Exp.  Zool.  iii,  pp.  337-3o7,  1906;  also  various  papers  by 
Max  Heidenhain,  in  Merkel  u.  Bonnet's  Anatomische  Hefte;   etc. 

X  These  motions  of  a  liquid  surface,  and  other  still  more  striking  movements, 
such  as  those  of  a  piece  of  camphor  floating  on  water,  were  at  one  time  ascribed 
by  certain  physicists  to  a  peculiar  force, ^sui  generis,  the  force  epipolique  of 
Dutrochet;  until  van  der  Mensbrugghe  shewed  that  differences  of  surface-tension 
were  enough  to  account  for  this  whole  series  of  phenomena  (Sur  la  tension  super- 
ficielle  des  liquides,  consideree  au  point  de  vue  de  certains  mouvements  observes 
a  leur  surface,  Mem.  Cour.  Acad,  de  Belgique,  xxxiv,  1869,  Phil.  Mag.  Sept.  1867; 
cf.  Plateau,  Statique  des  Liquides,  p.  283).  An  interesting  early  paper  is  by  Dr 
G.  Carradini  of  Pisa,  DelF  adesione  o  attrazione  di  superficie,  Mem.  di  Matem.  e 
di  Fisica  d.  Soc.  Ital.  d.  Sci.  (Modena),  xi,  p.  75,  xii,  p.  89,  1804-5. 


362  THE  FORMS  OF  CELLS  [ch. 

The  film  becomes  a  mere  bimolecular,  or  even  a  monomolecular, 
layer;  and  at  last  we  may  treat  it  as  a  simple  "surface  of  discon- 
tinuity." So  long  as  the  changes  due  to  imperfect  equihbrium  are 
taking  place  very  slowly,  we  speak  of  the  bubble  as  "at  rest";  it  is 
then,  as  Willard  Gibbs  remarks,  that  the  characters  of  a  film  are 
most  striking  and  most  sharply  defined*. 

So  also,  and  surely  not  less  than  the  soap-bubble,  is  every  cell- 
surface  a  complex  affair.  Face  and  interface  have  a  molecular 
orientation  of  their  own, -depending  both  on  the  partition-membrane 
and  on  the  phases  on  either  side.  It  is  a  variable  orientation, 
changing  at  short  intervals  of  space  and  time;  it  coincides  with 
inconstant  fields  of  force,  electrical  and  other;  it  initiates,  and 
controls  or  catalyses,  chemical  reactions  of  great  variety  and 
importance.  In  short  we  acknowledge  and  confess  that,  in  sim- 
pHfying  the  surface  phenomena  of  the  cell,  for  the  time  being  and 
for  our  purely  morphological  ends,  we  may  be  losing  sight,  or 
making  abstraction,  of  some  of  its  most  specific  physical  and 
physiological  characteristics. 

In  the  case  of  the  naked  protoplasmic  cell,  as  the  amoeboid  phase 
is  emphatically  a  phase  of  freedom  and  activity,  of  unstable  equi- 
librium, of  chemical  and  physiological  change,  so  on  the  other  hand 
does  the  spherical  form  indicate  a  phase  of  stabihty,  of  inactivity, 
of  rest.  In  the  one  phase  we  see  unequal  surface-tensions  manifested 
in  the  creeping  movements  of  the  amoeboid  body,  in  the  rounding- 
off  of  the  ends  of  its  pseudopodia,  in  the  flowing  out  of  its  substance 
over  a  particle  of  "food,"  and  in  the  current-motions  in  the  interior 
of  its  mass ;  till,  in  the  alternate  phase,  when  internal  homogeneity 
and  equilibrium  have  been  as  far  as  possible  attained  and  the 
potential  energy  of  the  system  is  at  a  minimum,  the  cell  assumes  a 
rounded  or  spherical  form,  passes  into  a  state  of  "rest,"  and  (for  a 
reason  which  we  shall  presently  consider)  becomes  at  the  same  time 
encysted  f. 

*  On  the  equilibrium  of  heterogeneous  substances,  Collected  Works,  i,  pp.  55-353; 
Trans.  Conn.  Acad.  1876-78. 

f  We  still  speak  of  the  naked  protoplasm  of  Amoeba;  but  short,  and  far  short, 
of  "encystment,"  there  is  always  a  certain  tendency  towards  adsorptive  action, 
leading  to  a  surface-layer,  or  "plasma- membrane,"  still  semi-fluid  but  less  fluid  than 
before,  and  different  from  the  protoplasm  within ;  it  was  one  of  the  first  and  chief 
things  revealed  by  the  new  technique  of  "micro-dissection."     Little  is  known  of 


v]  THE  FORM  OF  AMOEBA  363 

In  their  amoeboid  phase  the  various  Amoebae  are  just  so  many 
varying  distributions  of  surface-energy,  and  varying  amounts  of 
surface-potential*.  An  ordinary  floating  drop  is  a  figure  of  equi- 
librium under  conditions  of  which  we  shall  soon  have  something  to 
say;  and  if  both  it  and  the  fluid  in  which  it  floats  be  homogeneous 
it  will  be  a  round  drop,  a  "figure  of  revolution."  But  the  least 
chemical  heterogeneity  will  cause  the  surface-tension  to  vary  here 
and  there,  and  the  drop  to  change  its  form  accordingly.  The  httle 
swarm-spores  of  many  algae  lose  their  flagella  as  they  settle  down, 
and  become  mere  drops  of  protoplasm  for  the  time  being;  they 
"put  out  pseudopodia" — in  other  words  their  outline  changes;  and 
presently  this  amoeboid  outHne  grows  out  into  characteristic  lobes 
or  lappets,  a  sign  of  more  or  less  symmetrical  heterogeneity  in  the 
cell-substance. 

In  a  budding  yeast-cell  (Fig.  103  A),  we  see  a  definite  and  restricted 
change  of  surface-tension.  When  a  "bud"  appears,  whether  with 
or  without  actual  growth  by  osmosis  or  otherwise 
of  the  mass,  it  does  so  because  at  a  certain  part 
of  the  cell-surface  the  tension  has  diminished,  and 
the  area  of  that  portion  expands  accordingly ;  but 
in  turn  the  surface-tension  of  the  expanded  or  ex- 
truded portion  makes  itself  felt,  and  the  bud 
rounds  itself  off  into  a  more  or  less  spherical  form.  ^^' 

The  yeast-cell  with  its  bud  is  a  simple  example  of  an  important 
principle.  Our  whole  treatment  of  cell-form  in  relation  to  surface- 
tension  depends  on  the  fact  (which  Errera  was  the  first  to  give  clear 
expression  to)  that  the  incipient  cell-wall  retains  with  but  little 
impairment  the  properties  of  a  liquid  filmf,  and  that  the  growing 
cell,  in  spite  of  the  wall  by  which  it  has  begun  to  be  surrounded, 

the  physical  nature  of  this  so-called  membrane.  It  behaved  more  or  less  like  a  fluid 
lipoid  envelope,  immiscible  with  its  surroundings.  It  is  easily  injured  and  easily 
repaired,  and  the  well-being  of  the  internal  protoplasm  is  said  to  depend  on  the 
maintenance  of  its  integrity.  Robert  Chambers,  Physical  Properties  of  Protoplasm, 
1926;  The  living  cell  as  revealed  by  microdissection,  Harvey  Lectures,  Ser.  xxii, 
1926-27;   Journ.  Gen.  Physiol,  vm,  p.  369,  1926;   etc. 

*  See  (int.  al.)  Mary  J.  Hogue,  The  effect  of  media  of  different  densities  on  the 
shape  of  Amoebae,  Journ.  Exp.  Zool.  xxii,  pp.  565-572,  1917.  Scheel  had  said 
in  1889  that  A.  radiosa  is  only  an  early  stage  of  ^.  proteus  {Festschr.  z.  70.  Geburtstag 
0.  V.  Kuptfer). 

t  Cf.  infra,  p.  561. 


364  THE  FORMS  OF  CELLS  [ch. 

behaves  very  much  Hke  a  fluid  drop.  So,  to  a  first  approximation, 
even  the  yeast-cell  shews,  by  its  ovoid  and  non-spherical  form,  that 
it  has  acquired  its  shape  under  some  influence  other  than  the  uniform 
and  symmetrical  surface-tension  which  makes  a  soap-bubble  into  a 
sphere.  This  oval  or  any  other  asymmetrical  form,  once  acquired, 
may  be  retained  by  virtue  of  the  solidification  and  consequent 
rigidity  of  the  membrane-like  wall  of  the  cell;  and,  unless  rigidity 
ensue,  it  is*  plain  that  such  a  conformation  as  that  of  the  yeast-cell 
with  its  attached  bud  could  not  be  long  retained  as  a  figure  of  even 
partial  equilibrium.  But  as  a  matter  of  fact,  the  cell  in  this  case 
is  not  in  equilibrium  at  all ;  it  is  in  process  of  budding,  and  is  slowly 
altering  its  shape  by  rounding  off  its  bud.  In  like  manner  the 
developing  egg,  through  all  its  successive  phases  of  form,  is  never 
in  complete  equilibrium:  but  is  constantly  responding  to  slowly 
changing  conditions,  by  phases  of  partial,  transitory,  unstable  and 
conditional  equihbrium. 

There  are  innumerable  solitary  plant-cells,  and  unicellular 
organisms  in  general,  which,  hke  the  yeast-cell,  do  not  correspond 
to  any  of  the  simple  forms  which  may  be  generated  under  the 
influence  of  simple  and  homogeneous  surface-tension ;  and  in  many 
cases  these  forms,  which  we  should  expect  to  be  unstable  and 
transitory,  have  become  fixed  and  stable  by  reason  of  some  com- 
paratively sudden  solidification  of  the  envelope.  This  is  the  case, 
for  instance,  in  the  more  comphcated  forms  of  diatoms  or  of  desmids, 
where  we  are  dealing,  in  a  less  striking  but  even  more  curious  way 
than  in  the  budding  yeast-cell,  not  with  one  simple  act  of  formation, 
but  with  a  complicated  result  of  successive  stages  of  localised  growth, 
interrupted  by  phases  of  partial  consolidation.  The  original  cell 
has  acquired  a  certain  form,  and  then,  under  altering  conditions 
and  new  distributions  of  energy,  has  thickened  here  or  weakened 
there,  and  has  grown  out,  or  tended  (as  it  were)  tc  branch,  at  par- 
ticular points.  We  can  often  trace  in  each  particular  stage  of 
growth,  or  at  each  particular  temporary  growing  point,  the  laws  of 
surface  tension  manifesting  themselves  in  what  is  for  the  time  being 
a  fluid  surface ;  nay  more,  even  in  the  adult  and  completed  structure 
we  have  little  difficulty  in  tracing  and  recognising  (for  instance  in 
the  outline  of  such  a  desmid  as  Euastrum)  the  rounded  lobes  which 
have  successively  grown  or  flowed  out  from  the  original  rounded  and 


V]  OF  LIQUID  FILMS  365 

flattened  cell.  What  we  see  in  a  many  chambered  foraminifer,  such 
as  Glohigerina  or  Rotalia,  is  the  same  thing,  save  that  the  stages  are 
more  separate  and  distinct,  and  the  whole  is  carried  out  to  greater 
completeness  and  perfection.  The  little  organism  as  a  whole  is  not 
a  figure  of  equihbrium  nor  of  minimal  area;  but  each  new  bud  or 
separate  chamber  is  such  a  figure,  conditioned  by  the  forces  of 
surface-tension,  and  superposed  upon  the  complex  aggregate  of 
similar  bubbles  after  these  latter  have  become  consoHdated  one  by 
one  into  a  rigid  system. 

Let  us  now  make  some  enquiry  into  the  forms  which  a  fluid 
surface  can  assume  under  the  mere  influence  of  surface-tension. 
In  doing  so  we  are  limited  to  conditions  under  which  other  forces 
are  relatively  unimportant,  that  is  to  say  where  the  surface  energy 
is  a  considerable  fraction  of  the  whole  energy  of  the  system;  and 
in  general  this  will  be  the  case  when  we  are  dealing  with  portions 
of  liquid  so  small  that  their  dimensions  come  within  or  near  to  what 
we  have  called  the  molecular  range,  or,  more  generally,  in  which 
the  "specific  surface"  is  large.  In  other  words  it  is  the  small  or 
minute  organisms,  or  small  cellular  elements  of  larger  organisms, 
whose  forms  will  be  governed  by  surface-tension;  while  the  forms 
of  the  larger  organisms  are  due  to  other  and  non-molecular  forces. 
A  large  surface  of  water  sets  itself  level  because  here  gravity  is 
predominant;  but  the  surface  of  water  in  a  narrow  tube  is  curved, 
for  the  reason  that  we  are  here  deahng  with  particles  which  he  within 
the  range  of  each  other's  molecular  forces.  The  like  is  the  case  with 
the  cell-surfaces  and  cell-partitions  which  we  are  about  to  study,  and 
the  effect  of  gravity  will  be  especially  counteracted  and  concealed 
when  the  object  is  immersed  in  a  hquid  of  nearly  its  own  density. 

We  have  already  learned,  as  a  fundamental  law  of  "capillarity," 
that  a  liquid  film  in  equilibrium  assumes  a  form  which  gives  it  a 
minimal  area  under  the  conditions  to  which  it  is  subject.  These 
conditions  include  (1)  the  form  of  the  boundary,  if  such  exist,  and 
(2)  the  pressure,  if  any,  to  which  the  film  is  subject:  which  pressure 
is  closely  related  to  the  volume  of  air,  or  of  liquid,  that  the  film 
(if  it  be  a  closed  one)  may  have  to  contain.  In  the  simplest  of  ca